What can baby lizards do that baby snakes can’t

The angle of elevation of the sun is approximately 71.56 degrees when a 42-ft mast casts a 15-ft shadow.

We can use trigonometry to determine the angle of elevation of the sun. Let's call the angle we're looking for "theta".

We know that the tangent of theta is equal to the opposite side (the height of the mast) divided by the adjacent side (the length of the shadow):

tan(theta) = height of mast / length of shadow

tan(theta) = 42 / 15

Now we can use a calculator to find the inverse tangent of both sides of the equation:

theta = [tex]tan^{-1}(42/15)[/tex]

theta ≈ 71.56 degrees

So the angle of elevation of the sun is approximately 71.56 degrees.

If you want to verify the solution, you can use the following diagram to better understand the problem and see how the angle of elevation is related to the height of the mast and the length of the shadow:

In this diagram, the angle theta represents the angle of elevation of the sun, the height of the mast is 42 ft, and the length of the shadow is 15 ft. We can see that the tangent of theta is equal to the height of the mast divided by the length of the shadow, which is what we used in our calculation:

tan(theta) = height of mast / length of shadow

tan(theta) = 42 / 15

We can then solve for theta using inverse tangent:

theta = [tex]tan^{-1}(42/15)[/tex]

theta ≈ 71.56 degrees

As a result, the sun is elevated at a about 71.56 degree angle.

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The area of the deck is 816 square feet.

We need to subtract the area of the pool from the area of the deck plus pool combination.

The dimensions of the deck plus pool combination can be found by adding twice the deck width to the pool dimensions.

The length of the deck plus pool combination is:

40ft + 2(6ft) = 52ft

The width of the deck plus pool combination is:

16ft + 2(6ft) = 28ft

Therefore, the area of the deck plus pool combination is:

52ft * 28ft = 1456ft²

The area of the pool is:

16ft * 40ft = 640ft²

So the area of the deck is:

1456ft² - 640ft² = 816ft²

Therefore, the area of the deck is 816 square feet.

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To the nearest foot, the car travels approximately 21 feet over an angle of π/3 radians.

In this problem, we are given an angle of π/3 radians. To find out how far the car travels, we need to calculate the length of the arc that the car travels along. The formula for the length of an arc is given by:

arc length = radius x angle in radians

So, in this case, the arc length that the car travels is:

arc length = 20 x π/3

arc length ≈ 20 x 1.05

arc length ≈ 21

Therefore, the solution of arc length is 21.

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The distance from Georgetown to Brett's house, measured in a straight line, is approximately 64.75 miles, which has been calculated through Pythagorean Theorem.

You may determine the right angled triangle's missing length using the Pythagorean Theorem. The triangle has three sides: the adjacent, which doesn't touch the hypotenuse, the opposite, which is always the longest, and the hypotenuse.

We can solve this question through the Pythagorean theorem. Let's assume that Brett's house is at point B, Springfield is at point S, and Georgetown is at point G  

We want to find the length of the line segment BG, which is the distance from Brett's house to Georgetown.

We can say that the length of the line segment BS is x miles (we don't know the value of x yet), and the length of the line segment SG is 17 miles. We can also say that the line segments BS and SG are perpendicular to each other, since Brett's house is due west of Springfield and due south of Georgetown.

Using the Pythagorean theorem, we can write:

[tex]BG^2 = BS^2 + SG^2[/tex]

Substituting the known values, we get:

[tex]BG^2 = x^2 + 17^2[/tex]

Simplifying and solving for BG, we get:

[tex]BG = sqrt(x^2 + 17^2)[/tex]

We also know that the line segments BS and SG form a right triangle, so we can use the Pythagorean theorem again to write:

[tex]x^2 + BG^2 = (17 + BG)^2[/tex]

Expanding and simplifying, we get:

[tex]x^2 + BG^2 = 289 + 34BG + BG^2[/tex]

Substituting BG^2 with its value from the first equation, we get:

[tex]x^2 + x^2 + 17^2 = 289 + 34BG + x^2 + 17^2[/tex]

Simplifying, we get:

[tex]2x^2 = 289 + 34BG[/tex]

Substituting BG with its value from the first equation, we get:

[tex]2x^2 = 289 + 34sqrt(x^2 + 17^2)[/tex]

Simplifying and solving for x, we get:

[tex]x = sqrt((289/2)^2 - 17^2) = sqrt(4196.25) = 64.75[/tex]

Therefore, the distance from Georgetown to Brett's house, measured in a straight line, is approximately 64.75 miles.

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The figure that has the same area as the parallelogram is (d) the rectangle with a length 40.2 and width 21.8

The area of a parallelogram is:

Area = base * perpendicular height

From the image,

base = 40.2, height = 21.8

Hence:

Area of parallelogram = 40.2 * 21.8

Area  = 876.36 unit²

The rectangle (D) with a length 40.2 and width 21.8, have an area:

Area of rectangle = length * width

Area = 40.2 * 21.8

Area = 876.36 unit²

The rectangle with a length 40.2 and width 21.8, have the same area as the parallelogram

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If 42 boys study German Of the students who study German, the percentage of boys studying German is approximately 48.48%.

Given that there are 270 students in Year 7, we can use the information provided to find the number of students studying each language.

Let the number of students studying French be 2x, then the number of students studying Spanish is 7x/3.

Hence, the number of students studying German is (270 - 2x - 7x/3).

We are also given that 42 boys study German. Let the number of boys studying German be y, then the number of girls studying German is (42 - y). We can set up an equation to represent this:

y/(42 - y) = x/(7x/3)

Simplifying this equation, we get:

3y = 14x - 2xy

Now, we can substitute (270 - 2x - 7x/3) for (14x) to obtain:

3y = (270 - 2x - 7x/3)*2/3 - 2xy

Simplifying this expression, we get:

9y = 540 - 4x - 14x - 6xy

9y = 540 - 18x - 6xy

We know that y = 42, so we can substitute that value in and solve for x:

9(42) = 540 - 18x - 6x(42)

378 = 540 - 330x

330x = 162

x = 0.49

Thus, the number of students studying French is approximately 0.98 (2x) and the number of students studying Spanish is approximately 1.36 (7x/3).

Now we can find the percentage of boys studying German by dividing the number of boys (42) by the total number of students studying German:

Percentage of boys studying German = (42/((270 - 2x - 7x/3))*100

Percentage of boys studying German = (42/(86.67))*100

Percentage of boys studying German = 48.48%

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Complete question is:

There are 270 students in Year 7. Each student studies one of French or German or Spanish. Of these 270 students, 2 study French, the number who study French : the number who study Spanish = 3:7. 42 boys study German Of the students who study German, what percentage are boys?

You must show your working.

Use the formula A = one-half b h to find the area of the base. The area of the base, A, is One-half (8) (14) = 56 meters. The volume of the prism, V is (56) (7) = 392 meters cubed.

A prism is a solid figure having two bases that are the same in size and shape, flat sides, and a continuous cross-section. The two bases of a prism, which might be triangles, rectangles, or any other polygon, determine the name of the prism in question. Prisms having triangle bases, for instance, are referred to as triangular prisms, whereas those with square bases are referred to as square prisms, and so on. Three rectangle lateral sides and two triangular bases make up a triangular prism.

Given that the figure is a right triangular prism with, base 8 meters and 14 m, also the height of the prism is 7 m.

Thus, the correct steps to calculate the volume of the right triangular prism are:

Use the formula A = one-half b h to find the area of the base.

The area of the base, A, is One-half (8) (14) = 56 meters.

The volume of the prism, V is (56) (7) = 392 meters cubed.

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The blue line represents the curve of the function that is being integrated. The green line represents the area under the curve between the lower and upper limits of integration. The yellow line represents the area between the two points on the curve. The red dot is the lower limit of integration, and the yellow dot is the upper limit of integration.

Integration is a way of finding the area under a function. The lower limit of integration is the point at which the area starts to be measured, while the upper limit of integration is the point at which the area stops being measured.

When calculating the area under the curve between two points, the lower and upper limits of integration can be identified by the red and yellow dots.

To calculate the area between the two points, we will use the formula for integration. This involves taking the integral of the function between the lower and upper limits of integration. This is done by summing the area of each small slice of the function between the two points.

Once the area under the curve is found, it can be compared to the area represented by the green line. If the green line is larger than the area under the curve, then the area under the curve will be negative. If the area under the curve is larger than the green line, then the area under the curve will be positive.

By comparing the green line to the area under the curve, it is possible to determine whether the area is positive or negative. This can help to solve mathematical problems that require integration.

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Answer: The probability statement that would yield an acceptable approximation is 1-P(X>25).

To see why, we first need to check if the conditions for using the Normal approximation to the binomial distribution are met. These conditions are:

The number of trials is large (n ≥ 10)

The probability of success for each trial is approximately the same (p is roughly constant)

The number of successes and failures are both at least 5 (np ≥ 5 and n(1-p) ≥ 5)

In this case, we have n = 150 and p = 0.1, so np = 15 and n(1-p) = 135, which satisfy the conditions.

To use the Normal approximation, we need to standardize the binomial distribution. We have:

mean = np = 15

standard deviation = sqrt(np(1-p)) = sqrt(13.5)

To find P(X>25), we can standardize X and use the Normal distribution:

Z = (25 - 15) / sqrt(13.5) = 3.06

P(X>25) = P(Z > 3.06) ≈ 0.0011

This means that the probability of more than 25 students in a class of 150 being left-handed is approximately 0.0011.

Therefore, the probability statement that yields an acceptable approximation is 1-P(X>25).

Step-by-step explanation:

The probability statement P(X<25) would yield an acceptable approximation, as it corresponds to the number of left-handed students being less than 25 out of the total of 150 students.

The chance of success raised to the power of the number of triumphs and the probability of failure raised to the power of the difference between the number of successes and the number of attempts are multiplied to create the binomial distribution.

For instance, if we flip a coin, there are only two potential results: heads or tails, and if we take a test, there are only two possible outcomes: pass or fail. A binomial chance distribution is another name for this distribution.

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The highest degree οf any term in the pοlynοmial is 13, which means that the degree οf the pοlynοmial is 13.

The degree οf a term in a pοlynοmial is the sum οf the expοnents οf the variables in that term.

The degree οf the pοlynοmial w³x²-14x⁹v²w² +5−2v can be fοund by determining the highest degree οf any term in the pοlynοmial.

The degree οf the first term w³x² is 3+2 = 5

The degree οf the secοnd term -14x⁹v²w² is 9+2+2 = 13

The degree οf the third term 5 is 0

The degree οf the fοurth term -2v is 1

Therefοre, the highest degree οf any term in the pοlynοmial is 13, which means that the degree οf the pοlynοmial is 13.

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Answer:

Maybe like 10-15 minutes away

Answer:

Theoretical probability for each color is 1/4, or 25%.

Experimental probability of blue is 42/200, or 21%.

Experimental probability of purple is 55/200, or 27.5%.

Experimental probability of green is 71/200, or 35.5%.

Experimental probability of red is 32/200, or 16%.

Correct statements:

Experimental property of purple (27.5%) is more than theoretical property of blue (25%).

Theoretical property of blue (25%) is more than experimental property of red (16%).

The required probability is P(T > 16.8) = e^(-λ*16.8) = e^(-16.8/1) = e^(-16.8) = 3.22*10^(-8).Hence, the required probability is e^(-16.8).

Given that priority boarding for a scheduled flight is announced to start, the time it takes for a random passenger with priority boarding to line up is described by an exponential probability distribution with expectation 1/lambda minutes. We assume that the time two random passengers spend queuing is independent of each other. It should be 56 passengers on a small scheduled flight to the north, 5 of whom have prioritized boarding.

The time it takes from ordinary boarding being announced to a random passenger without priority boarding queuing is described by the same probability distribution as above. We still assume that the time two random passengers spend queuing is independent of each other.

We are supposed to find the probability that it takes more than 16.8 minutes from ordinary boarding has started until all ordinary passengers have lined up for boarding. Express the answer as a function of λ. Solution: We can apply the Poisson distribution to calculate the probability that all ordinary passengers have lined up for boarding. This is because Poisson distribution models the number of arrivals in a given period of time, given the average arrival rate, λ.The number of passengers who are waiting at any given moment follows a Poisson distribution with an expected value of λ, the rate parameter.If the time it takes for a passenger to get into the line is exponential with an expectation of 1/λ minutes, then λ passengers arrive every minute.

Hence, the time between the arrivals of two passengers is exponential with a mean of 1/λ minutes, which implies that the probability density function (pdf) of a single time duration between two consecutive arrivals is:$$f_{T}(t)=\lambda e^{-\lambda t}, \ \ t \in [0,\infty)$$For any fixed t, the probability that it takes more than t time units to get a passenger in the queue is obtained by integrating the pdf over the corresponding interval, i.e. $$P(T>t)=\int_{t}^{\infty}\lambda e^{-\lambda u}du=e^{-\lambda t}, \ \ t \in [0,\infty)$$

Therefore, the probability that it takes more than T seconds to get a passenger in the queue is given by P(T>t) = e^(-λT), where T is the time in seconds.We need to find the probability that it takes more than 16.8 minutes from ordinary boarding has started until all ordinary passengers have lined up for boarding. Let the random variable T denote the time from when ordinary boarding was announced until the last ordinary passenger queued.

So, the required probability is P(T > 16.8) = e^(-λ*16.8) = e^(-16.8/1) = e^(-16.8) = 3.22*10^(-8).Hence, the required probability is e^(-16.8).

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One coterminal angle to 30 degrees is -330 degrees. 150 degrees and 510 degrees are coterminal angles. Similarly, any angle that differs by a multiple of 360 degrees from the given angle is coterminal with it.

Coterminal angles are angles that have the same initial side and the same terminal side. They can be either negative or positive, and they can have any degree measure. Coterminal angles are determined by adding or subtracting 360 degrees (or 2π radians) from the given angle as many times as necessary until the resulting angle falls between 0 and 360 degrees.

Here are the steps to determine coterminal angles: Add or subtract 360 degrees from the given angle. Keep adding or subtracting 360 degrees as many times as necessary to obtain an angle between 0 and 360 degrees. For example, consider the given angle of 150 degrees.150 degrees + 360 degrees = 510 degrees510 degrees - 360 degrees = 150 degreesThe angle 510 degrees is coterminal with 150 degrees. However, it is a very large angle that is difficult to visualize on the coordinate plane.

Therefore, we can find the smallest positive coterminal angle to 150 degrees by subtracting 360 degrees from 510 degrees as many times as necessary until we get an angle between 0 and 360 degrees.510 degrees - 360 degrees = 150 degrees (already between 0 and 360 degrees). Any angle that differs by a multiple of 360 degrees from the given angle is coterminal with it. For example, 870 degrees, 1230 degrees, and -330 degrees are all coterminal with 150 degrees.

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The standard form of the equation of the circle is (x - 1)² + (y + 3)² = 50, h=1, k= -3, r=  5√(2).

The standard form equation for a circle is:

(x - h)² + (y - k)² = r²

where (h, k) is the center of the circle and r is the radius. The equation represents all points (x,y) that are a fixed distance r from the center (h,k) of the circle.

To understand this equation, it may be helpful to visualize a circle on a coordinate plane. The center of the circle is located at the point (h, k), which is the midpoint of the circle. The radius of the circle is represented by r, which is the distance from the center of the circle to any point on the circumference of the circle.

The center of the circle is the midpoint of the diameter, which can be found using the midpoint formula:

Midpoint = ((x1 + x2)/2, (y1 + y2)/2)

where (x1, y1) and (x2, y2) are the endpoints of the diameter.

So, the midpoint is:

(((-4) + 6)/2, (-8 + 2)/2) = (1, -3)

This means that the center of the circle is (h, k) = (1, -3).

To find the radius r, we can use the distance formula between the center and one of the endpoints of the diameter:

r = √((x1 - h)² + (y1 - k)²)

Using (-4,-8) as one endpoint, we get:

r = √((-4 - 1)² + (-8 - (-3))²) = √(25 + 25) = 5√(2)

So the standard form of the equation of the circle is:

(x - 1)² + (y + 3)² = 50

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The dot represents the vertex, and the x represents the point where the parabola touches the x-axis. The parabola is symmetric about the vertical line x = h and does not cross the x-axis anywhere else.

A parabola is a type of curve that is defined by a specific mathematical equation, namely, the quadratic equation. It is a symmetrical curve that can be described as the shape of the graph of a quadratic function.

by the question.

If (h, k) is a zero of the function, then. [tex]f(h) = 0[/tex]. Substituting this into the equation for f(x), we get:

[tex]0 = a(h - h)^2[/tex]

[tex]0 = 0[/tex]

This is a true statement, which tells us that (h, k) is indeed a zero of the function.

Now, let's consider the graph of this function. Since the coefficient a is non-zero, the parabola will be facing either upwards or downwards. If a > 0, then the parabola will be facing upwards, and if a < 0, then the parabola will be facing downwards.

Since the vertex of the parabola is at (h, k), the axis of symmetry is the vertical line x = h. Therefore, the parabola is symmetric about this line.

Finally, since (h, k) is also a zero of the function, the parabola must cross the x-axis at x = h with a single point of tangency. This means that the parabola just touches the x-axis at this point and does not cross it.

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After answering the presented question, we can conclude that As a cylinder result, the depth of the water in the tank rises by 6.33 cm.

A cylinder is a three-dimensional geometric shape made up of two parallel congruent circular bases and a curving surface connecting the two bases. The bases of a cylinder are always perpendicular to its axis, which is an imaginary straight line passing through the centre of both bases. The volume of a cylinder is equal to the product of its base area and height. A cylinder's volume is computed as V = r2h, where "V" represents the volume, "r" represents the radius of the base, and "h" represents the height of the cylinder.

This cylinder has the following volume:

V_cylinder = π × r² × h

= π × (40 cm)² × (50 cm)

= 251,327.41 cm³

Hence the volume of water displaced by the cube is 125,000 cm, and the volume of water displaced by the cylinder with the same height and radius as the tank is 251,327.41 cm3. As a result, the depth of the water rises by:

Δh = V_cube / (π × r²) - V_cylinder / (π × r²)

= (125,000 cm³) / (π × (40 cm)²) - (251,327.41 cm³) / (π × (40 cm)²)

= 6.33 cm

As a result, the depth of the water in the tank rises by 6.33 cm.

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The total number of students is calculated by dividing the total number of chess club sign ups by the proportion of students who signed up Therefore, 210/0.6 = 840 students in the school.

15 students out of 25 signed up for chess club.

Therefore, 15/25 = 0.6 of the students signed up for chess club.

Therefore, 0.6 x total number of students = 210 (the number of students who signed up for chess club).

Therefore, 210/0.6

= 840 students in the school.

In one class, 15 out of 25 students signed up for the chess club. This means that 0.6 (or 60%) of the students in the class decided to join. If this rate stays constant across the entire school, then it is possible to calculate the total number of students in the school by using the proportion of students who signed up for the chess club. To calculate the total number of students, we use the equation: 0.6 x total number of students = 210 (the number of students who signed up for chess club). By dividing both sides of the equation by 0.6, we get 210/0.6 = 840 students in the school. Therefore, there are 840 students in the whole school if 210 signed up for chess club.

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Therefore, the surface area of the cylinder is approximately 301m².

In mathematics, area refers to the measurement of the size of a two-dimensional surface or region. It is a measure of the amount of space inside a flat figure, such as a square, circle, or triangle. The area of a figure is usually expressed in square units, such as square centimeters, square meters, or square inches.

Here,

The net of the right circular cylinder consists of three rectangles: one for the lateral surface, and two for the top and bottom faces.

The lateral surface of a cylinder can be found using the formula:

Lateral surface area = 2πrh

where π is the value of pi (approximately 3.14), r is the radius of the base, and h is the height of the cylinder. In this case, the height of the cylinder is given as 8m, and the radius is given as 4m (half of the width of one of the rectangles). Therefore, we have:

Lateral surface area = 2πrh = 2(3.14)(4m)(8m) ≈ 201m²

The area of each of the circular faces is given by the formula:

Circular face area = πr²

Since the radius is 4m, we have:

Circular face area = πr² = 3.14(4m)² ≈ 50m²

The total surface area of the cylinder is the sum of the lateral surface area and the areas of the two circular faces:

Total surface area = Lateral surface area + 2 × Circular face area

Total surface area = 201m² + 2(50m²) = 301m² (rounded to the nearest whole number)

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Answer:

[tex]125cm^{3}[/tex]

Step-by-step explanation:

TSA of a Cuboid=  2LW x 2LH x 2WH
250 = 2 ( L X W X H)
250 / 2 = L X W X H
125 = L X W X H

The volume of a cuboid = L X W X H
125 = L X W X H
Volume = 125cm3

Answer:

Step-by-step explanation:

Answer:



Step-by-step explanation:

Given: The table of linear function.

x :  -3  , 0 , 3,  6

y :  -6 , -2 , 2,  6

Slope=change in y over change in x



Passing point: (0,-2)

Point slope form:







Slope of the linear function

Linear function is

Hence, The slope of linear function is

The slope of the given linear function which graph represents the data is 3/2.

The slope of a linear function can be calculated by finding the difference between two points on the graph and dividing it by the difference of the corresponding x-values of those points.

In this case, the two points are (2, -2) and (6, 4).

The difference between these y-values = 6,

and the difference between the x-values = 4.

Therefore, the slope of the linear function which graph represents the data = 6/4

= 3/2

This means that the linear function has a slope of 3/2 which summarizes that for every two units that x increases, y increases by three units.

If x increases from 0 to 4, y increases from -5 to 1, which is a difference of 6 units (4 x 3/2 = 6).

The linear function can be written as y = 3/2x -5.

This means that for any given x-value, the corresponding y-value can be calculated by multiplying 3/2 by the x-value and subtracting 5.

If x = 2, the y-value is -2,that is

(2* 3/2)- 5= -2

Therefore, the slope of the linear function which graph represents the data is 3/2.

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After addressing the issue at hand, we can state that The parallelogram's dimensions are 6.54 in and 3.06 in to the nearest hundredth of an inch.

In Euclidean geometry, a parallelogram is a straightforward quadrilateral with two sets of parallel sides. Both sets of opposite sides in a parallelogram are parallel and equal. There are four types of parallelograms, three of which are unique. The four distinct shapes are parallelograms, squares, rectangles, and rhombuses. A quadrilateral becomes a parallelogram when it has two sets of parallel sides. The opposing sides and angles of a parallelogram are the same length. The interior angles are additional angles on the same side of the horizontal line. The sum of all interior angles is 360 degrees.

cos 40 = adjacent /hypotenuse base

0,7660 = 4 adjacent

adjoining = 4 x 0.7660 = 3.06 in

Area / base = length

20 / 3.06 = 6.54 in

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The correct question is -

The parallelogram has an area of 20 square inches. What are the dimensions of the parallelogram, to the nearest hundredth of an inch?

40°

4 in

h

OA. I = 2.57 in, h = 7.78 in

OB. I = 6.22 in, h = 3.23 in

The volume of the rectangular prism is 105.57

What is the volume of a rectangular prism?

A rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces.

The volume of a rectangular prism=Length X Width X Height

Where l, h and w represent the length, height and width respectively.

Given that;

Length of the prism is 2 times the width L = 2w

the width is 4 times the height  w = 4h

Now let’s plug these values;

the surface area equation, it becomes;

A = 2(2w(w) + w(w/4) + w/4(2w))

A = 2(2w^2 + w^2/4 + w^2/2)

A = 1/2(7w^2)

792 x2 = 7w^2

w = 15.04

The volume of a rectangular prism=Length X Width X Height

V = 2w x w x w/4

V x 2 = w³

V = 15.04³/2

V = 105.57

Thus, the volume is 105.57.

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The probability that the three marbles selected are all different colors is 0.224 or 22.4%.

To find the probability that the three marbles selected are all different colors from a box that contains 4 green marbles, 9 blue marbles, and 6 red marbles, one at a time, without replacement. We can use the probability formula as follows:P(all different colors) = Number of ways to choose 3 marbles of different colors / Total number of ways to choose 3 marbles

The total number of ways to choose 3 marbles from a total of 19 marbles in the box is given by:19C3 = (19 × 18 × 17) / (3 × 2 × 1) = 969The number of ways to choose 3 marbles of different colors from a box containing 4 green marbles, 9 blue marbles, and 6 red marbles, one at a time, without replacement can be obtained as follows:4C1 × 9C1 × 6C1 = (4 × 9 × 6) = 216

Therefore, the probability of choosing 3 marbles of different colors from the box is:P(all different colors) = Number of ways to choose 3 marbles of different colors / Total number of ways to choose 3 marblesP(all different colors) = 216/969P(all different colors) = 24/107 or 0.224 or 22.4%

Therefore, the probability that the three marbles selected are all different colors is 0.224 or 22.4%.

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According to the given conditions of volume,[tex]$V_{cone} = \frac{1}{3} \pi \frac{120}{\pi h} h = \frac{1}{3} \cdot 120 = 40}$[/tex] cubic inches.

Volume is the amount of space that a three-dimensional object occupies or contains. It is a measure of the total amount of enclosed space inside a solid figure, such as a cube, cylinder, sphere, or any other three-dimensional shape.

According to given information :

Since the cylinder and cone have the same height and congruent circular bases, their volumes are proportional to the squares of their radii.

Let the radius of the base of the cylinder and cone be denoted as r.

The volume of a cylinder is given by:

[tex]$V_{cylinder} = \pi r^2 h$[/tex]

where h is the height of the cylinder.

We are given that the volume of the cylinder is 120 in, so we can plug this into the formula and solve for r:

[tex]$120 = \pi r^2 h$[/tex][tex]$r^2 = \frac{120}{\pi h}$[/tex]

The volume of a cone is given by:

[tex]$V_{cone} = \frac{1}{3} \pi r^2 h$[/tex]

We know that the cone and cylinder have the same height, so we can substitute the expression we found for [tex]$r^2$[/tex] into the formula for the volume of the cone:

According to the given conditions,

[tex]$V_{cone} = \frac{1}{3} \pi \frac{120}{\pi h} h = \frac{1}{3} \cdot 120 = 40}$[/tex] cubic inches.

Therefore, the volume of the cone is 40 cubic inches.

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Isaac can fit 27 boxes inside the carton.

To determine how many boxes will fit inside the carton, we need to calculate the volume of each box and the volume of the carton. The volume of each box is 4 x 6 x 2 = 48 cubic inches. The volume of the carton is 12 x 18 x 6 = 1296 cubic inches. To find the number of boxes that will fit, we divide the volume of the carton by the volume of each box: 1296 / 48 = 27. Therefore, 27 boxes can fit inside the carton. It's important to note that this calculation assumes the boxes will be packed in a way that utilizes all available space without any wasted gaps.

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[tex]$\cos 6x$[/tex] can be expressed in terms of the first power of cosine as [tex]$2(\cos^2 x)^3 - 1$[/tex].

Structure-based algebraic expressiοn rewriting is the same as rearranging an expressiοn tο plug it intο anοther expressiοn. Sοlve fοr οne οf the variables in these kinds οf prοblems and then insert the resulting expressiοn fοr that variable intο the οther expressiοn.

We may rewrite [tex]$cos 6x$[/tex]  in terms of [tex]$cos 2 x$[/tex] and [tex]$cos 4 x$[/tex] using the fοrmula for lowering powers as fοllows:

[tex]\begin{aligned}\cos 6x = \cos^2 3x - \sin^2 3x \\= \cos^2 3x - (1 - \cos^2 3x) \\= 2\cos^2 3x - 1 \\= 2(\cos^2 x)^3 - 1. \end{aligned}[/tex]

Hence, [tex]$\cos 6x$[/tex]can be written in terms οf the first pοwer of cosine as [tex]$2(\cos^2 x)^3 - 1$[/tex].

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The value of scale factor is 18/35 and the segment PT when ST is parallel to QR is 7.71 cm.

Triangles that are similar to one another in terms of shape but differ in terms of size. In other words, the corresponding angles and sides of two comparable triangles are equal and proportionate.

Because the ratios of the respective sides of comparable triangles are identical, they may be used to solve proportional and scaling issues. As a result, if we are aware of the ratio of two sides in one triangle, we may utilise the similarity of the triangles to determine the ratio of the other triangle.

The scale factor is:

scale factor = PT/QP = (54/7)/15 = 18/35

Using similar triangles, we can set up the following proportion:

PT/PR = SQ/QR

Substituting the given values, we get:

PT/15 = 6/(6+PT)

PT = 54/7 cm

Therefore, the value of scale factor is 18/35 and the segment PT when ST is parallel to QR is 7.71 cm.

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