Lorene plans to make several open-topped boxes in which to carry plants. She makes the boxes from rectangular sheets of cardboard from
which she cuts out 4 in squares from each corner. The length of the original piece of cardboard is 8 in more than the width. If the volume of
the box is 420 in", determine the dimensions of the original piece of cardboard.
The dimensions of the original piece of cardboard are __ in by _ in.
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[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{D varies directly with G}}{D = k(G)}\hspace{5em}\textit{we also know that} \begin{cases} G=24\\ D=552 \end{cases} \\\\\\ 552=k(24)\implies \cfrac{552}{24}=k\implies 23=k\hspace{5em}\boxed{D=23G}[/tex]

Inequality to find the value of p is 7p - 12 ≤ 45. So correct option is C.  The value of p is at most 8.14 centimeters.

Inequalities can be solved in a similar way to equations, but with some important differences. To solve an inequality, one must find the set of values that satisfy the inequality. This set of values is often expressed as an interval, which is a range of values between two endpoints. For example, the solution to the inequality x < 5 is the interval (-∞, 5), which includes all values of x that are less than 5. The solution to an inequality may also be expressed graphically on a number line or coordinate plane.

The perimeter of an isosceles triangle with sides of length 3p-6, 3p-6, and p is:

Perimeter = (3p-6) + (3p-6) + p

= 7p - 12

We are given that the perimeter is at most 45 centimeters. Therefore, we can write the following inequality to find the value of p:

7p - 12 ≤ 45

Adding 12 to both sides, we get:

7p ≤ 57

Dividing both sides by 7, we get:

p ≤ 8.14

Therefore, the value of p is at most 8.14 centimeters.

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The group of rotations and symmetries of a geometric configuration is the group of isometries.

According to the above paragraph, "the group of rotations and symmetries of the configuration" refers to the group of isometries of a geometric configuration that preserve the configuration itself, without involving any translations. This group consists of rotations about a fixed point and symmetries about a straight line that pass through that point.

This group is formed by combining isometries of the configuration through the operation of composition, and it satisfies certain properties, such as closure, associativity, identity, and inverses. It is a useful tool for studying the symmetries and transformations of the configuration, and it is used in fields such as crystallography and quantum mechanics.

In summary, the group of rotations and symmetries of a geometric configuration is the group of isometries that preserve the configuration and can be composed with each other to obtain new isometries.

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a. To find the speed at which the car achieves maximum fuel economy, we need to find the vertex of the function. The vertex of a quadratic function is given by the formula: (-b/2a, f(-b/2a)). In this case, a = -0.008 and b = 1.4. Substituting these values, we get:

Vertex = (-b/2a, f(-b/2a))

= (-1.4/(2*(-0.008)), f(1.4/(2*(-0.008))))

= (87.5, 6.125)

Therefore, the car should travel at a speed of 87.5 miles per hour to achieve maximum fuel economy.

b. To find the maximum fuel economy, we need to find the value of the function at the vertex. Substituting x = 87.5 in the function, we get:

f(87.5) = -0.008(87.5)^2 + 1.4(87.5) = 6.125

Therefore, the maximum fuel economy of the car is 6.125 miles per gallon.

c. To find the fuel economy of the car if it travels at 60 miles per hour, we need to evaluate the function at x = 60. Substituting x = 60 in the function, we get:

f(60) = -0.008(60)^2 + 1.4(60) = 7.6

Therefore, the fuel economy of the car if it travels at 60 miles per hour is 7.6 miles per gallon.

d. The point (40, 43.2) lies on the graph of the function. This means that when the car travels at a speed of 40 miles per hour, its fuel economy is 43.2 miles per gallon. However, this point does not provide any information about the maximum fuel economy or the speed at which it is achieved.

The values of the measures are;

Range = 6

Median = 4

Mean = 4.5

To determine the values of the range, median and mean, we need to note that;

The range of a given set of values is the interval between the smallest number and the largest number in the set.

Given the data set;

3, 4, 5, 7, 8, 8, 9

The range = 9 - 3 = 6

The median of a given set of values is the  middle number of the set when arranged in an ascending or descending form.

Given the set;

3, 4, 4, 6, 6,

The median is 4

The mean of a set of numbers is the average value of the numbers.

Given the set;

2, 9, 7, 5, 2, 2,

The mean = 2 + 9 + 7 + 5 + 2 + 2/6 = 27/6 = 4. 5

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In response to the given question, we can state that Because it is independent of z, the function g(z) is a constant function. In particular, g(z) equals 2 - 1 + 1 = 2.

Mathematicians examine numbers and complex variations, equations and associated structures, forms and their locations, and prospective positions for these things. The term "functioning" signifies the connection between a collection of inputs, each of which has a corresponding output. A function is a connection of inputs and results in which each input leads to a single, identifiable outcome. Each function is assigned a domain, a codomain, or a scope. The letter f is widely used this to denote functions (x). The symbol for admission is an x. The four primary types of usable functions are on operations, one-to-one capabilities, so multiple functionality, in capabilities, and then on functions.

Because it is independent of z, the function g(z) is a constant function. In particular, g(z) equals 2 - 1 + 1 = 2.

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Answer:

Step-by-step explanation:

[tex]\tt z + 2 < 8[/tex]

Subtract 2 from both sides:-

Simplify :-

Therefore, the solution of the inequality z + 2 < 8 is z < 6.

_______________________

Hope this helps!

The intervals for which the rate of change of the quadratic function, f(x) = -(x + 1)² + 8 is faster than the rate of change of g(x) = -x - 6 are; x ≤ -1.5 and x ≥ -0.5

A quadratic function is a polynomial function with one or more variables in which the highest exponent is two.

The slope of f(x) is; f'(x) = -2·(x + 1)

The slope of g(x) = -1

When the rate of change of f(x) is faster than the rate of change of g(x), we get;

The slope of f(x) > The slope of g(x), therefore;

-2·(x + 1) > -1

From which we get;

|-2·(x + 1)| ≥ |-1|

2·(x + 1) ≥ 1

x + 1 ≥ 1/2

x ≥ 1/2 - 1 = -1/2

x ≥ -1/2 = -0.5

The absolute sign indicates that we get;

x + 1 ≤ -1/2

x ≤ -1/2 - 1 = -1 -1/2 = -3/2 = -1.5

x ≤ -1.5

The interval for which the average rate of change of f(x) is faster than the average rate of change of g(x) are; x ≥ -0.5 and x ≤ -1.5

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The probability of having 4 hurricanes in a year is approximately 0.118 or 11.8%.

It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

a. To find the probability of having exactly 4 hurricanes in a year, we use the Poisson distribution with a mean of 6.6:

P(X=4) = (e^-6.6)×(6.6^4)/(4!) ≈ 0.118

Therefore, the probability of having 4 hurricanes in a year is approximately 0.118 or 11.8%.

b. In a 35-year period, the expected number of years with 4 hurricanes is:

μ = λt = 6.635 ≈ 231

Therefore, we can expect to have 231 years with 4 hurricanes in a 35-year period.

c. If in a recent period of 35 years, only 3 years had 4 hurricanes, this is less than what the Poisson distribution would predict (231 years). However, this doesn't necessarily mean that the Poisson distribution doesn't work well here. The Poisson distribution is a theoretical model that assumes certain conditions, and it's possible that those conditions weren't met in this specific case. It's also possible that this was just a rare event that can happen due to chance. Further analysis would be needed to determine whether the Poisson distribution is a good fit for this data.

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Answer:

e = 10

Step-by-step explanation:

Vertically opposite angles are the angles formed opposite to each other when two lines intersect.

Vertically opposite angles are equal.

          9e - 8 = 8e + 2

Add 8 to both sides,

                 9e = 8e + 2 + 8

                 9e = 8e + 10

Subtract '8e' from both sides,

         9e - 8e = 10

                   [tex]\boxed{\bf e = 10}[/tex]

Answer:

e = 10

Step-by-step explanation:

According to the Vertical Angle Theorem, when two straight lines intersect, the opposite vertical angles are congruent.

Since the two given angles are vertical angles, they are congruent.

Therefore, we can set them equal to each other:

[tex]\implies (9e - 8)^{\circ} = (8e + 2)^{\circ}[/tex]

[tex]\implies 9e - 8= 8e + 2[/tex]

To solve for e, begin by subtracting 8e from both sides of the equation:

[tex]\implies 9e - 8-8e= 8e + 2-8e[/tex]

[tex]\implies e - 8= 2[/tex]

Add 8 to both sides of the equation:

[tex]\implies e - 8+8= 2+8[/tex]

[tex]\implies e=10[/tex]

Therefore, the value of e is 10.

As you can see, each number in the pattern is obtained by adding 4 to the previous number.

An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.

Here,

Sure, here's a pattern for the rule a + 4:

a = 1: 1 + 4 = 5

a = 2: 2 + 4 = 6

a = 3: 3 + 4 = 7

a = 4: 4 + 4 = 8

a = 5: 5 + 4 = 9

a = 6: 6 + 4 = 10

a = 7: 7 + 4 = 11

a = 8: 8 + 4 = 12

a = 9: 9 + 4 = 13

a = 10: 10 + 4 = 14

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Answer:

NO= 18

Step-by-step explanation:

If we dissect the image, we can see that triangle LMN there are two similar triangles; triangle LMN and smaller triangle OPN. By definition of similar triangles, corresponding parts of similar triangles will have the same ratios. So, as we can see, sides NP and NM are corresponding. The ratio of the two sides is 24/44 (just put the smaller side over the longer side). Simplified, the ratio is 6/11. So, we know that the opposite sides' ratio will also be 6/11. So, all we have to do is multiply NL (33) by 6/11, which equals 18.

Check:

24/44=6/11, and 18/33=6/11, so the ratios are the same.

The correct statement regarding the numeric values of the piece-wise function is given as follows:

D. f(1) = 0.

To obtain the numeric value of a function or of an expression, we substitute each instance of the variable in the function or in the expression by the value at which we want to find the numeric value.

A piece-wise function is a function that has different definitions based on the input of the function, hence before obtaining the numeric value we must obtain the correct definition of the function in the interval.

Choosing the correct definitions of the function, the numeric values are given as follows:

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Answer:

simplified expression: 15yz + 12

The proportional and inverse relationships can be presented as follows;

1. The value of y when x = 36 is 6

2. The value of y when x = 1 is 27

3. The time of fall of an object that falls 144 feet is 3 seconds

4. The length of the string that vibrates 64 times per second is 48 inches long

A proportional relationship is one that can be expressed in the form; y = k × x.

The possible questions, obtained from a similar question posted online, are presented as follows;

1. y varies directly as the square root of x, when y = 4, x = 16, to find y when x = 36;

The value of y can be found by expressing the relationship between y and x using a proportional relation as follows;

y ∝ √x

y = c·√x

4 = c·√(16) = 4·c

Therefore;

c = 4/4 = 1

c = 1

When x = 36, we get;

y = 1 × √(36) = 6

2. The variation of y and x can be presented as follows;

y ∝ 1/x³

y = c/x³

y = 1, when x = 3, therefore;

1 = c/3³ = c/27

c = 27 × 1 = 27

c = 27

The value y when x = 1, can be found as follows;

y = 27/1³

y = 27/1³ =27

3. The relationship between the distance and the time duration the ball can be presented as follows;

Let s represent the distance the ball falls, and let t represent the time the ball falls, we get;

s = k·t²

The distance an object falls in one second = 16 feet

Therefore, we get;

When s = 16, t = 1

16 = k × 1²

k = 16/1² = 16

k = 16

s = 16·t²

When s = 144 feet, we get;

144 = 16 × t²

t² = 144/16 = 9

t = √9 = 3

4. The relationship between the vibration of a string and the tension in the string can be presented as follows;

Let f represent the vibration of the string, and let l represent the length in the string, we get;

f ∝ 1/l

f = c/l

When l = 24, f = 128, we get;

128 = c/24

c = 128 × 24 = 3,072

Therefore, if the string vibrates 64 times per second, we get;

64 = 3,072/l

l = 3,072/64 = 48

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Answer:

The sum of the exterior angles of any polygon, regardless of the number of sides, is always 360 degrees. Each exterior angle of a regular octagon measures 45 degrees, because the sum of the interior and exterior angles at each vertex is 180 degrees, and the octagon has 8 vertices. Therefore, the sum of all the exterior angles of an octagon is:

8 × 45º = 360º

So the answer is option B) 360º.

The distance οn the map that represents οne mile is 5/8 inch.

Tο find οut hοw much distance οn the map represents οne mile, we need tο use a ratiο οf the distance οn the map tο the actual distance.

Let x be the distance οn the map that represents οne mile. Then we have:

1/2 inch : 4/5 miles = x : 1 mile

We can crοss-multiply tο sοlve fοr x:

(1/2 inch) × (1 mile) = (4/5 miles) × x

Simplifying the right-hand side:

(1/2) × 1 = (4/5) × x

1/2 = (4/5) × x

Multiplying bοth sides by 5/4:

(1/2) × (5/4) = x

5/8 = x

Therefοre, the distance οn the map that represents οne mile is 5/8 inch.

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In response to the question, we may say that  Value total: 60 cents plus equation 240 cents plus 300 cents equals $3.00.

The equals symbol (=), which indicates equivalence, connects two statements in a mathematical equation. An algebraic equation's mathematical assertion proves the equality of two mathematical propositions. The equal sign, for instance, places a gap between the numbers 3x + 5 and 14 in the equation 3x + 5 = 14. Use a mathematical formula to understand how the two sentences on opposite sides of a letter relate to one another. Usually, the logo corresponds to the particular programme. An example would be 2x - 4 = 2.

Let's define our variables first:

Let x represent the amount of quarters in the machine.

10x + 5(8x) = 300

When we simplify and find x, we obtain:

10x + 40x = 300

50x = 300\sx = 6

Hence, the machine has six dime coins. We may determine the quantity of nickels by using the connection between the number of dimes and nickels:

8x = 8(6) = 48

There are 48 nickels in the machine as a result.

Verifying our response:

6 dime is 6 times 10 cents, or 60 cents.

48 nickels is 48 x 5 cents, or 240 cents.

Value total: 60 cents plus 240 cents plus 300 cents equals $3.00.

The vending machine has 48 nickels and 6 dimes, thus our answer is right.

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Answer:

93%

Step-by-step explanation:

651/700 houses were for sale. Write that as a percentage.

651/700=0.93

0.93=93%

Answer:

(1,2)

Step-by-step explanation:

The solution to a system of equations is the point at which the lines defined by the equations intersect.

On this graph, we can see that the lines meet a point on the graph that is at 1 on the x-axis (horizontal axis) and at 2 on the y-axis (vertical axis).

In Cartesian coordinates, this is written in the format (x, y):

(1, 2)

Answer:

x = -5y+9/4

Step-by-step explanation:

4x + 5y = 9

↓

(4x + 5y) + (-5y) = 9 + (-5y)

↓

4x + 5y - 5y = 9 - 5y

↓

4x = -5y + 9

↓

4x/4 = -5y + 9/4

↓

x = -5y + 9/4

Answer: x = 9/4 - 5y/4

Step-by-step explanation: Move all terms that don't contain x to the right side and solve.

Step-by-step explanation:

Divide.

Answer: 680

Step-by-step explanation:

Answer:

5, 8 and 2

Because it has the same angel as 3 there for it's 5 , 8 and 2

A rectangle has a width that is 16 feet shorter than its length and an area that is 35 square feet. The rectangle's length and breadth can be calculated using the equation [tex]3x^2[/tex] - 16x - 35 = 0.

Assume that the rectangle measures "x" feet in length. Then, according to the problem:

The width is 16 feet less than 3 times the length, so the width is 3x - 16 feet.

The area of the rectangle is 35 square feet, so we can write:

Area = Length x Width

35 = x(3x - 16)

To solve for x, we can simplify this quadratic equation by expanding the right-hand side and moving all the terms to one side:

35 = [tex]3x^2[/tex] - 16x

[tex]3x^2 - 16x - 35 = 0[/tex]

The rectangle's length and breadth can be calculated using the quadratic equation shown above.

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Answer:

The rocket splashes down after [tex]\boxed{24}[/tex] seconds

The rocket peaks at [tex]\boxed{769}[/tex] meters above sea-leval

Step-by-step explanation:

The relationship between height and time is given by the equation
[tex]h(t) = -4.9t^2+112t+129[/tex]

The splashdown will occur when [tex]h(t) = 0[/tex]

Therefore solve t for
[tex]-4.9t^2+112t+129 = 0[/tex]

This is a quadratic equation which can be solved with the aid of a quadratic formula calculator

The solution set for this equation is
[tex]t=-1.09894 \;and\;\:t=23.95609\right)[/tex]

Omitting the negative value we get
[tex]\text{t=23.95609 \;seconds}\\\\= 2\text{4 \;seconds \;rounded}}[/tex]

For the second part, the maximum value of h(t) can be found by finding the first derivative h'(t) and setting it equal to 0 and solving for t; then plug this value of t into h(t) to find the max height

[tex]h'(t) = \dfrac{d}{dt}\left(-4.9t^2+112t+129\right)\\\\= \dfrac{d}{dt}\left(4.9t^2\right)+\dfrac{d}{dt}\left(112t\right)+\dfrac{d}{dt}\left(129\right)\\\\= - 9.8t + 112 + 0\\\\= -9.8t + 112\\\\[/tex]

Setting  the above expression to 0 gives
[tex]-9.8t+112=0\\\\-9.8t = -112\\\\t = \dfrac{-112}{-9.8}\\\\t = 11.4286 \;seconds[/tex]

Plugging this value back into h(t) gives

[tex]h(11.4286) = -4.9\cdot \:11.4286^2+112\cdot \:11.4286+129\\\\= 769 \;meters \;(rounded)[/tex]

The triangle with side lengths 6, 9, and 12 is an obtuse triangle. To classify the triangle as acute, right, or obtuse, we need to determine the type of triangle based on the lengths of its sides.

Recall that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side for it to be a valid triangle.

Let's check if the given triangle with side lengths 6, 9, and 12 satisfies the triangle inequality:

1. Sum of two shorter sides: 6 + 9 = 15 (greater than the length of the longest side, 12). Valid.

2. Sum of the two longer sides: 9 + 12 = 21 (greater than the length of the remaining side, 6). Valid.

3. Sum of the longest side and the shortest side: 6 + 12 = 18 (greater than the length of the remaining side, 9). Valid.

Since all three inequalities are satisfied, the given side lengths form a valid triangle.

Now, to classify the triangle, we can use the Pythagorean theorem. In a right-angled triangle, the square of the length of the longest side is equal to the sum of the squares of the other two sides.

In this case, the side lengths are 6, 9, and 12. To determine if it is a right-angled triangle, we can check if:

[tex]12^2 = 6^2 + 9^2\\144 = 36 + 81\\144 \neq 117[/tex]

Since the Pythagorean theorem is not satisfied, the triangle is not a right-angled triangle.

Now, let's check if it is an acute or obtuse triangle. In an acute triangle, all three angles are less than 90 degrees, while in an obtuse triangle, one angle is greater than 90 degrees.

To determine this, we can use the law of cosines, which states that in any triangle:

[tex]c^2 = a^2 + b^2 - 2ab * cos(C)[/tex]

where c is the longest side (12 in this case), a and b are the other two sides (6 and 9), and C is the angle opposite the longest side.

Calculate the value of cos(C):

[tex]12^2 = 6^2 + 9^2 - 2 * 6 * 9 * cos(C)\\144 = 36 + 81 - 108 * cos(C)\\144 = 117 - 108 * cos(C)\\108 * cos(C) = 117 - 144\\108 * cos(C) = -27\\cos(C) = -27 / 108\\cos(C) = -0.25[/tex]

Since the value of cos(C) is negative, the angle C is obtuse. Therefore, the triangle is an obtuse triangle.

In summary, the triangle with side lengths 6, 9, and 12 is an obtuse triangle.

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In a casde whereby saac is mountain climbing with Rina and has just climbed a 6.2-meter vertical rock face the distance from the cliff is Rina standing is 7.85m

Base on the given iformation, following the trigonometry rule it can be seen thatr the hypotenus of the triangle is the distance of Isaac  to Rina which can be expressed a C = 10m

The height that was climbed by I saac =  6.2-m. a= 6.2m

the distance from Rina to cliff = b

Then base on pytagoras theorem

c^2= a^2 + b^2

10^2 = 6.2^2 +  b^2

b^2= 10^2 - 6.2^2

b^2= 61.56

b = √61.56

b=7.85m

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Answer:

-9 and -5

Step-by-step explanation:

-9 + -5 = -14

-9 * -5 = 45

Answer:

- 9 and - 5

Step-by-step explanation:

- 9 × - 5 = 45 and - 9 + (- 5) = - 9 - 5 = - 14

Step-by-step explanation:

it's radius squared timesed by pi (π)

r^2×π

if u don't have a radius but a diameter then half the diameter to get the radius

Answer:

There are two common formulas that can be used to calculate the area of a circle. Here they are:

Where [tex]\pi[/tex] is the universal value for all circles that expresses the ratio between the length of circumference and the diameter of any circle. It's a rational number and its value is about 3.141592653589793238462643383279502884197..., so you may want to just 3.14.

This is the easiest and simplest formula. Letter "r" represents the length of radius of the circle,

Rarely used formula. In this equation, "C" represents the length of circumference of the circle.

Check the attached image to better understand the meaning of radius, diameter and circumference. Also, provided a second image with all the formulas for the area of a circle.