Use the terms from the list below to complete the sentences.
Each term will be used once.
constant of proportionality
proportional relationship
+
ratio
table
equivalent ratios
equation
It takes a machine 10 seconds to fill 4 bottles of juice. It takes 15 seconds
to fill 6 bottles of juice. Does this describe a proportional relationship?
1. Make a(n)
to organize the data.
Time (t)
10.
15
Number of
Bottles (b)
4
6

The intermediate step in the form (x+a)²=b as a result of completing the square is: (x + 7)² = -4. We can calculate it in the following manner.

To complete the square for the equation x² + 22x = 8x - 53, we need to move the constant term to the right side and the linear term to the left side, and then add a constant value to both sides so that the left side becomes a perfect square trinomial. The steps are as follows:

x² + 22x = 8x - 53 (original equation)

x² + 14x = -53 (subtract 8x from both sides)

x² + 14x + (14/2)² = -53 + (14/2)² (add (14/2)² to both sides)

x² + 14x + 49 = -53 + 49 (simplify)

(x + 7)² = -4 (factor the left side and simplify the right side)

So the intermediate step in the form (x+a)²=b as a result of completing the square is:

(x + 7)² = -4

Note that this equation has no real solutions, since the square of any real number is always nonnegative, whereas the right side of this equation is negative.

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The value of x:z = 1:4.

To find the ratio of x to z, we need to have a common term between x, y, and z. Since we have y in both ratios, we can use it as the common term.

From the given ratios:

x:y = 3:4

y:z = 1:3

We can see that the y in the first ratio and the y in the second ratio are the same.

To find the ratio of x to z, we can "connect" the two ratios by canceling out the y:

x:y = 3:4

y:z = 1:3

x:y × y:z = 3:4 × 1:3

x:z = 3:12

Simplifying the above ratio by dividing both terms by 3, we get:

x:z = 1:4

It would cost $288.00 to recarpet the shop with carpet that costs $3.00 per square meter.

An expression contains one or more terms with addition, subtraction, multiplication, and division.

We always combine the like terms in an expression when we simplify.

We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.

Example:

1 + 3x + 4y = 7 is an expression.com

3 + 4 is an expression.

2 x 4 + 6 x 7 – 9 is an expression.

33 + 77 – 88 is an expression.

We have,

The area of the shop can be found by multiplying its length by its width:

Area = Length x Width

        = 8 meters x 12 meters

        = 96 square meters

To recarpet the entire shop, we need to purchase 96 square meters of carpet.

If the carpet costs = $3.00 per square meter.

The total cost can be found by multiplying the area by the cost per square meter:

Total cost = Area x Cost per square meter

Total cost = 96 square meters x $3.00 per square meter

Total cost = $288.00

Therefore,

It would cost $288.00 to recarpet the shop with carpet that costs $3.00 per square meter.

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The equation [tex]24 + n = 39[/tex] may be used to get n, the quantity of scarves Rebecca purchases.

The concept of an equation in algebra is a logical statement that demonstrates the equality of two mathematical equations. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

A formula that describes how contain a copy on either sides of a symbol are connected. It typically has an equal sign and one variable. The variable x is included in the preceding example.

The cost of one scarf is $[tex]5[/tex]

The cost of one purse is $[tex]12[/tex]

Rebecca buys 2 purses, so the cost of the purses is [tex]2*12=24[/tex]

The total cost of the purchase is $[tex]39[/tex]

[tex]Total cost = Cost of scarves + Cost of purses[/tex]

[tex]39 = 5n + 24[/tex]

Now we can solve n,

[tex]39 - 24 = 5n[/tex]

[tex]15 = 5n[/tex]

[tex]n = \frac{15}{5}[/tex]

[tex]n=3[/tex]

Therefore, Rebecca buys [tex]3[/tex] scarves.

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Answer:

(-1,0) and (0, 4)

Step-by-step explanation:

The x and y intercept of a function (say the x intercept), is when y = 0 and vice versa.

The x intercept in this function is when y = 0

Thus,

4x-0=-4

4x = -4

x = -1.

So the x- intercept point = (-1, 0)

Similarly,

4(0)-y= -4

(x is equated to 0 to find the y - intercept)

-y = -4

∴ y = 4

y- intercept point = (0, 4)

Hope this helps! :)

The evaluation of the statements and the values of the side lengths and angles of the quadrilateral are;

A. 1.) [tex]\overline{BD}[/tex] ≅ [tex]\overline{AC}[/tex]; False

2.) [tex]\overline{BO}[/tex] ⊥ [tex]\overline{CO}[/tex]; True

3. ∠BDA ≅ ∠BDC; False

4. ∠BOA ≅ ∠CBD; False

5.) m∠BAD + m∠ADC = 180°; True

6.) [tex]\overline{CO}[/tex] ≅ [tex]\overline{AO}[/tex]; True

7.) ∠BCD = 90°; False

8.) ∠BOA = 90°; True

9.) ∠ABD = 45°; False'

10. [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex]; True

B. I. ∠NAL = 124°

∠CNL = 28°

∠CLN = 28°

∠NLA = 28°

II. 5.) ∠1 = 45°

∠2 = 90°

III 7.) [tex]\overline{OP}[/tex] = 7 cm

8.) [tex]\overline{OE}[/tex] = 12 cm

9.) [tex]\overline{EO}[/tex] = 13 cm

10. ∠5 = 60°

A quadrilateral is a four sided polygon.

The evaluation of the quadrilaterals are presented as follows;

1. [tex]\overline{BD}[/tex] ≅ [tex]\overline{AC}[/tex]

[tex]\overline{BD}[/tex] and [tex]\overline{AC}[/tex] are the diagonals of the rhombus, and the properties of a rhombus indicates that the diagonals of a rhombus are not congruent, the statement,  [tex]\overline{BD}[/tex] ≅ [tex]\overline{AC}[/tex] is therefore false

[tex]\overline{BD}[/tex] ≅ [tex]\overline{AC}[/tex]; False

2.  [tex]\overline{BO}[/tex] ⊥ [tex]\overline{CO}[/tex]

[tex]\overline{BO}[/tex] and [tex]\overline{CO}[/tex] are segments on the two diagonals of the rhombus which are perpendicular. The statement, [tex]\overline{BO}[/tex] ⊥ [tex]\overline{CO}[/tex] is therefore; True

3. ∠BDA ≅ ∠BDC

∠ADC = ∠BDA + ∠BDC

The diagonals of a rhombus do not bisect the vertex angles, therefore;

m∠BDA ≠ m∠BDC

∠BDA [tex]\ncong[/tex] ∠BDC

The statement, ∠BDA ≅ ∠BDC, therefore is; False

4.) ∠BOA ≅ ∠CBD

The diagonals of a rhombus are perpendicular, therefore angle ∠BOA = 90°

∠CBD is an interior angle of the rhombus, and ∠CBD = 90° if the rhombus is a square.

The specified rhombus is not shaped like a square. The statement, ∠BOA ≅ ∠CBD, is therefore; False

5.) m∠BAD + m∠ADC = 180°

The angles, ∠BAD and ∠ADC are adjacent interior angles, therefore, according to the properties of a rhombus, m∠BAD + m∠ADC = 180°. The statement is therefore; True

6.) [tex]\overline{CO}[/tex] ≅ [tex]\overline{AO}[/tex]

The segments [tex]\overline{CO}[/tex] and [tex]\overline{AO}[/tex] are formed by the intersection of the diagonal [tex]\overline{BD}[/tex] and [tex]\overline{AC}[/tex] at O. The diagonals of a rhombus bisect each other. The statement, [tex]\overline{CO}[/tex] ≅ [tex]\overline{AO}[/tex] is therefore; True

7.) ∠BCD = 90°

∠BCD = 90° when the rhombus is a square, the statement is therefore False

8.) ∠BOA = 90°

∠BOA is formed by the intersection of the diagonals of a rhombus, which are perpendicular to each other. Therefore, ∠BOA = 90°, the statement is therefore; True

9) ∠ABO = 45°

∠ABO = 45° if the rhombus is a square, the statement is therefore; False or more information required

10) [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex]

[tex]\overline{BC}[/tex] and [tex]\overline{CD}[/tex] are side lengths of the rhombus and are therefore, congruent. The statement, [tex]\overline{BC}[/tex] ≅ [tex]\overline{CD}[/tex] is; True

B. I. ∠NAL and ∠CNA are supplementary, therefore;

∠NAL + ∠CNA = 180°

∠NAL= 180° - ∠CNA

∠CNA = 56°

∠NAL= 180° - 56° = 124°

2.) Whereby the diagonal of the rhombus bisects the angle ∠CNA, we get;

∠CNL = ∠ANL = 56°/2 = 28°

3.) ∠CLN = ∠CNL = 28°

4.) ∠NLA = ∠CNL = 28°

II. The properties of a square indicates that we get;

The diagonals of a square bisect the vertex angles, therefore;

∠1 which is the angle formed by the the diagonal [tex]\overline{AT}[/tex] that bisects the angle ∠CAR = 90° is; ∠1 = 90°/2 = 45°

6.) The diagonals of a square are perpendicular, therefore;

∠2 = 90°

III. 7.) The facing sides of a rectangle are congruent, therefore;

[tex]\overline{HE}[/tex] ≅ [tex]\overline{OP}[/tex]

[tex]\overline{OP}[/tex] = [tex]\overline{HE}[/tex] = 7 cm

8.) The diagonals of a square have the same length, therefore;

The length of [tex]\overline{OE}[/tex] = The length of [tex]\overline{HP}[/tex] = 12 cm

9.) Pythagorean Theorem indicates, that we get;

[tex]\overline{EO}[/tex] = √([tex]\overline{OP}[/tex]² + [tex]\overline{EP}[/tex]²)

Therefore;  [tex]\overline{EO}[/tex] = √(5² + 12²) = 13

[tex]\overline{EO}[/tex] = 13 cm

10.) ∠2 and ∠5 are vertical angles, therefore;

∠2 ≅ ∠5

∠5 ≅ ∠2 (Symmetric property)

m∠5 = m∠2 = 60° (Definition of congruency)

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For the rectangle, the answers will be a. Perimeter=52 inches, b. Perimeter=770 inches and c. Perimeter will be multiplied by 35 also.

What exactly is a rectangle?

A rectangle is a four-sided flat shape with opposite sides that are parallel and equal in length. It is a type of quadrilateral, a polygon with four sides.

Now,

(a) The perimeter of the drawing is the sum of the lengths of all sides of the rectangular garden plus the lengths of the two bricks on the top and bottom and the lengths of the two bricks on the left and right.

The length of the garden in the drawing is 7 inches, and the width is 4 inches. Thus, the perimeter of the garden in the drawing is:

P = 2(7 inches) + 2(4 inches) = 14 inches + 8 inches = 22 inches

Since the garden is surrounded by bricks, we need to add the lengths of the two bricks on the top and bottom and the lengths of the two bricks on the left and right. Each brick has a length of 1 inch, so the total length of the bricks is:

2(1 inch + 7 inches + 1 inch) + 2(1 inch + 4 inches + 1 inch) = 2(9 inches) + 2(6 inches) = 18 inches + 12 inches = 30 inches

Therefore, the perimeter of the drawing is:

22 inches + 30 inches = 52 inches

(b) The actual garden is 35 times larger than the drawing, so its length is 35 × 7 inches = 245 inches, and its width is 35 × 4 inches = 140 inches. Thus, the perimeter of the actual garden is:

P = 2(245 inches) + 2(140 inches) = 490 inches + 280 inches = 770 inches

(c) When the dimensions of the garden are multiplied by 35, the perimeter of the garden is also multiplied by 35. This is because the perimeter is a linear function of the length and width of the garden. Therefore, the effect on the perimeter of the garden is to increase it by a factor of 35.

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We found that he can only use approximately 7.39 units of fencing for the fourth side. The possible locations for point H are (4.5, 12.39) and (4.5, -2.39).

To determine where Alex could place point H, we need to first calculate the length of the three sides of the fenced area that he has already plotted. We can do this using the distance formula:

Distance [tex]= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

From point E to point F:

Distance = [tex]\sqrt{(6-3)^2 + (5-5)^2}[/tex]

               = 3 units

From point F to point G:

distance = [tex]\sqrt{(9-6)^2 + (1-5)^2)}[/tex]

              = [tex]\sqrt{(9 + 16)}[/tex]

              = 5 units

From point G to point E:

Distance = [tex]\sqrt{(3-9)^2 + (5-1)^2}[/tex]

                = [tex]\sqrt{(36 + 16)}[/tex]

                = [tex]\sqrt{(52)[/tex]

                = [tex]2 \sqrt{(13)}[/tex] units

Therefore, the total length of fencing that Alex has already plotted is 3 + 5 + 2√13 = approximately 14.61 units.

To find the possible x-coordinate(s) of point H, we can use the formula for the distance between two points on the coordinate plane:

distance = [tex]\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]

For point H to be 7.39 units away from the midpoint of side EF (4.5, 5), we have:

[tex]\sqrt{(x - 4.5)^2 + (y - 5)^2}[/tex]

= 7.39

Squaring both sides and simplifying gives:

(x - 4.5)² + (y - 5)²= 7.39²

Since we know that x=4.5, we can substitute this value into the equation and solve for y:

(4.5 - 4.5)²+ (y - 5)² = 7.39²

(y - 5)² = 7.39²

y - 5 = ±7.39

y = 5 ± 7.39

So the possible locations for point H are (4.5, 5+7.39) and (4.5, 5-7.39), which correspond to the points (4.5, 12.39)

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Charlotte, North Carolina had an approximate population of  531,145 in 2000.

To approximate Charlotte's population in 2000, we can use the formula for exponential growth:

P(t) = [tex]P0 \times e^{(rt)[/tex]

where P(t) is the population at time t, P0 is the initial population, r is the annual rate of growth as a decimal, and e is the mathematical constant e (approximately 2.71828).

Let's let t = 13 be the number of years between 2000 and 2013. We know that P(13) = 775,000, and r = 0.032. We can solve for P0 as follows:

775,000 = [tex]P0 \times e^{(0.03213)[/tex]

P0 = [tex]775,000 / e^{(0.03213)[/tex]

Using a calculator, we can approximate P0 as:

P0 ≈ 531,145

Therefore, an approximation of Charlotte's population in 2000 is  531,145.

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We might estimate that 0.8n is the approximate prediction number of times it won't land on the elephant.

We need to know the likelihood that the spinner will land on an elephant as well as the total number of spins in order to anticipate the number of times it won't.

Assume the spinner contains five equal parts, one of which is decorated with an elephant. The likelihood of the spinner touching the elephant is then 1/5, or 0.2.

If we know the total number of spins, we can use the likelihood that the spinner will fall on an elephant to calculate the likelihood that it won't. 1 - 0.2 = 0.8 is the probability that the spinner will miss the elephant.

Hence, if we suppose that the spinner is spun n times, we may estimate that 0.8n is the approximate number of times it won't land on the elephant. The prediction of probability of the spinner landing on the elephant is assumed to be constant for each spin and that each spin is independent of the others.

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The required perpendicular segment to DE is EF.

In simple geometry, two geometric objects are perpendicular if the intersection at the place of intersection known as a foot results in right angles. The perpendicular symbol can be used to graphically depict the condition of perpendicularity.

In the given diagram we can see that segment DE and EF.

So, we can say that,

Segment DE is perpendicular to segment EF.

Thus, required perpendicular segment to DE is EF.

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Complete question:

Answer:

3.87 miles

Step-by-step explanation:

The attached figure gives the relative locations of the courthouse, the library and the pool.

The three locations represent a right triangle with the distance between courthouse - pool as the hypotenuse and the distance between courthouse - library as the vertical leg

By the Pythagorean theorem if c is the hypotenuse and a, b the other two legs

[tex]c^2 = a^2 + b^2[/tex]

a = distance between courthouse and library = 7 miles
c = distance between courthouse  and pool = 8 miles
and we have to find b

So plugging in known values
[tex]8^2 = 7^2 + b^2\\\\b^2 = 8^2 - 7^2\\\\b^2 = 64 - 49\\\\b^2 = 15\\\\b= \sqrt{15} = 3.87298 \approx 3.87 \;miles[/tex]

Step-by-step explanation:

Volume of a cylinder = πr²h

[tex]\pi = 3.14[/tex]

[tex]r = \frac{d}{2} [/tex]

[tex]d = 14[/tex]

[tex]r = \frac{14}{2} [/tex]

[tex]r = 7[/tex]

[tex] {r}^{2} = 49[/tex]

[tex]h = 12[/tex]

substitute the formular with the values above

[tex]3.14 \times 7 \times7 \times 12 = 1846.32[/tex]

To one decimal place

[tex] = 1846.3[/tex]

[tex]1846.3[/tex]

If you buy three hamburgers, the cost of each hamburger is $2.50.

One hamburger costs $2.50.

To determine the price of one hamburger, first take the total cost of all three hamburgers, which is $7.50.

Then, divide the total cost of all three hamburgers by the number of hamburgers, which is three.

This yields $2.50, which is the price of one hamburger.

One of the four fundamental arithmetic operations, or how numbers are joined to create new numbers, is division.

The other operations are multiplication, addition, and subtraction.

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Answer:

22 rows

Step-by-step explanation:

Divide 48 by 2.25 and round up

The equation y = (-2/3)x + 6 represents a line that starts at the point (0, 6) on the y-axis and slopes downwards from left to right at a rate of 2 units down for every 3 units to the right.

Coordinate geometry is a branch of mathematics that deals with the study of geometry using the principles of algebra. It involves using algebraic equations and geometric concepts to analyze shapes and figures in a plane or in higher dimensions.

In this equation, -2/3 is the slope of the line, which tells us how steep the line is and in what direction it's heading. Specifically, a slope of -2/3 means that for every increase of 3 units in the x-direction, the y-value decreases by 2 units. So the line slopes downwards from left to right.

The 6 in the equation is the y-intercept of the line, which tells us where the line intersects the y-axis. Specifically, the y-intercept is the point (0, 6) on the line. This means that when x = 0, y = 6, so the line starts at the point (0, 6) on the y-axis.

Therefore, the equation y = (-2/3)x + 6 represents a line that starts at the point (0, 6) on the y-axis and slopes downwards from left to right at a rate of 2 units down for every 3 units to the right.

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Your speed is 60 miles per hour, when your friends average speed is 50 miles.

A rational function, which is a function that can be represented as the ratio of two polynomials, is a function that may be used in a rational model, which is a mathematical model. Rates of change, growth or decay, and proportions are only a few examples of the many various kinds of real-world events that may be represented using rational models. In order to describe complicated systems or processes, they are frequently employed in disciplines like economics, physics, and engineering. To determine the values of the variables that make the equation true, rational models can be solved using algebraic techniques including factoring, simplification, and cross-multiplication.

Given that, average speed is 10 miles per hour faster than other person.

Then, your speed is = s + 10.

For 180 miles, and 150 miles for friend the equation can be set as:

180/(s+10) = 150/s

We can cross-multiply to simplify:

180s = 150(s+10)

180s = 150s + 1500

30s = 1500

s = 50

Substituting the value in s + 10 = 50 + 10 = 60.

Hence. your speed is 60 miles per hour.

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Answer:

Let's denote the measures of the other two angles by 13x and 16x, where x is a constant.

Since the sum of the angles in a triangle is always 180 degrees, we can write an equation:

35 + 13x + 16x = 180

Simplifying this equation, we get:

29x = 145

x = 5

Therefore, the measures of the other two angles are:

13x = 13(5) = 65 degrees

16x = 16(5) = 80 degrees

So, the measures of the other two angles are 65 degrees and 80 degrees.

Step-by-step explanation:

Answer:

(0,-2)

Step-by-step explanation:

we can use the cubic equation formula to solve for t. The answer is t=1988, which means that there were 31.35 thousand European students in 1988.

To solve this problem, we need to find the value of t that corresponds to 31.35 thousand European students. We can use the given equation S=0.05(t3-11t+45t+277) and solve it for t. To do this, we can first subtract 277 from both sides to get S-277=0.05(t3-11t+45t). Then, we can divide both sides by 0.05 to get (S-277)/0.05=t3-11t+45t. Finally, we can use the cubic equation formula to solve for t. The answer is t=1988, which means that there were 31.35 thousand European students in 1988.

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Answer: 412.5 ml

Step-by-step explanation:

500 ( 1 - 25%) ( 1 + 10%)

= 412.5 ml

Answer:

412.5 ml

Step-by-step explanation:

To decrease 500 ml by 25%, you would need to multiply 500 by 0.25 to find the amount of decrease:

Subtract that amount from the original value:

Therefore decreasing 500 ml by 25% would give us 375 ml.

To then increase this value by 10%, you would multiply it by 0.10 to find the amount of increase:

Now, add that amount to the previous value:

Therefore, if you decrease 500 ml by 25% and then increase it by 10%, the final result is 412.5 ml.

On solving the question we have that Therefore, the center of the circle equation is (3/10, -7/10) and the radius is √(1/5).

A math equation is a mechanism for connecting two statements and indicating equivalence with the equals sign (=). To explain the connection between the two sentences put on each side of a letter, a statistical method can be employed. The software and the logo are usually interchangeable. 2x - 4 equals 2, for example. An equation is a logical expression that asserts the equality of some mathematical expressions in algebra. In the equation 3x + 5 = 14, for example, the equal sign separates the numbers 3x + 5 and 14.

The given equation is that of a circle in standard form:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

where the center of the circle is (h, k) and the radius is r.

To convert the given equation to this form, we need to complete the square for both x and y terms. Let's start with the x terms:

[tex]5x^2 - 3x = 5(x^2 - (3/5)x)\\5(x^2 - (3/5)x + (3/10)^2 - (3/10)^2)\\5((x - 3/10)^2 - 9/100)\\5y^2 + 7y = 5(y^2 + (7/5)y)\\5(y^2 + (7/5)y + (7/10)^2 - (7/10)^2)\\5((y + 7/10)^2 - 49/100)\\5((x - 3/10)^2 - 9/100 + (y + 7/10)^2 - 49/100) - 1 = 0\\5(x - 3/10)^2 + 5(y + 7/10)^2 = 1\\[/tex]

Therefore, the center of the circle is (3/10, -7/10) and the radius is √(1/5).

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Answer: C

Step-by-step explanation:

For A, look at the slope of the graph: 70/2 = 35

For B, slope calculation: m = y-y / x-x = 315-105 / 9-3 = 35

=> C

The cot(cos−1(a))= a/√(1-a^2).

Value of cos(sin−1(1)): Let's assume that θ= sin−1(1). It means that sinθ= 1. The value of θ is π/2, therefore we can say that cos(sin−1(1))=cos(π/2)= 0b) Value of tan(cos−1(-23)): Let's suppose that θ= cos−1(-23). It implies that cosθ= -23/25. As we know that tanθ= sinθ/cosθ, Therefore;tan(cos−1(−23))= sin(θ)cos(θ) = -24/25.2. a) Value of sin(cos−1(3/2)): Let's suppose that θ= cos−1(3/2). It implies that cosθ= 3/2. As we know that sin^2θ= 1- cos^2θ, Therefore;sin(cos−1(3/2))= √(1- (3/2)^2)= √(1/4)= 1/2.b) Value of csc(tan−1(-5/1)): Let's assume that θ= tan−1(-5/1). It implies that tanθ= -5/1. As we know that cscθ= 1/sinθ, Therefore;csc(tan−1(−51​))= 1/sin(θ)= 1/√(1+tan^2θ) = 1/√(1+25)= -1/√26.3. a) Value of sin(tan−1(a^2)): Let's suppose that θ= tan−1(a^2). It implies that tanθ= a^2. As we know that sinθ= tanθ/√(1+tan^2θ), Therefore;sin(tan−1(a2​))= (a^2)/√(1+(a^4)).b) Value of cot(cos−1(a)): Let's suppose that θ= cos−1(a). It implies that cosθ= a. As we know that cotθ= cosθ/sinθ, Therefore;cot(cos−1(a))= a/√(1-a^2).

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Answer:

percentage = (part/whole) x 100

where the "part" is the number of swimmers who finished in less than 39 minutes, and the "whole" is the total number of swimmers.

So, if 29 out of 50 swimmers finished in less than 39 minutes:

percentage = (29/50) x 100

percentage = 58

Therefore, 58% of swimmers finished the race in less than 39 minutes.

An equivalent fraction to represent 58% is 29/50.

Step-by-step explanation:

The number of integers between two given integers is 124.

The two given integers are 48 and 172.

We know that, number of integers between two integers = Difference of two integers

Here, number of integers = 172-48

= 124

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To optimise the volume while maintaining the total dimensions at 7 inches, the camera's dimensions should be roughly 3.01 inches besides 3.311 inches besides 0.679 inches.

What 3 dimensions do we have?

The homes we reside in and the items we use on a daily basis all have three dimensions: length, weigth, and breadth.

Let's start by assigning variables to the dimensions. Let x be the height of the camera, then the length must be 1.1 times the height, which gives us a length of 1.1x.

The width is not explicitly given, but we can express it in terms of x and 1.1x. Since the sum of the dimensions is 7 inches, we have:

x + 1.1x + w = 7

where w is the width of the camera. Simplifying this equation, we get:

2.1x + w = 7

w = 7 - 2.1x

Now we can express the volume of the camera in terms of x:

V = x(1.1x)(7 - 2.1x)

Simplifying this expression, we get:

V = 8.235x³ - 15.365x² + 7x

To find the maximum volume, we need to find the value of x that maximizes V. We can do this by taking the derivative of V with respect to x, and setting it equal to zero:

dV/dx = 24.705x² - 30.73x + 7 = 0

Using the quadratic formula, we can answer this quadratic equation:

x = (-b ± √(b² - 4ac)) / 2a

where a = 24.705, b = -30.73, and c = 7. Plugging in these values, we get:

x = 0.735 inches or x = 3.01 inches

Since x represents the height of the camera, we discard the smaller root and take x = 3.01 inches.

Then the length is 1.1 times the height, which gives us a length of 3.311 inches.

The width can be found using the equation w = 7 - 2.1x, which gives us w = 0.679 inches.

Therefore, the dimensions of the camera should be approximately 3.01 inches by 3.311 inches by 0.679 inches to maximize the volume while keeping the sum of the dimensions at 7 inches.

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Answer: The answer is 40

Explanation: The picture