If the sides of a square are increased by 11, the area becomes 400. What is the length of the original side?

The inverse function is:

f⁻¹(y) = √(y - 2), where y ≥ 2.

An inverse function is a function that "undoes" the action of another function. In other words, if we start with a value, apply a function to it, and then apply the inverse function to the result, we should get back to the original value. For example, if we have a function f(x) that doubles a number, the inverse function would be one that halves a number. The notation for an inverse function is f⁻¹(x), and it is defined as follows:

If f is a function with domain A and range B, then its inverse function f⁻¹         is a function with domain B and range A, where f⁻¹(y) = x if and only if f(x) = y.

To find the inverse of the function f(x) = x² + 2, we need to solve for x in terms of y.

Step 1: Replace f(x) with y.

y = x² + 2

Step 2: Solve for x in terms of y.

y - 2 = x²

±√(y - 2) = x

Note that we use ± because when we take the square root of a number, we get both a positive and a negative solution. However, since the domain of the function is x ≥ 0, we only consider the positive solution.

So the inverse function is:

f⁻¹(y) = √(y - 2), where y ≥ 2.

To sketch the graphs of f(x) and f¹(x) on the same coordinate plane:

Step 1: Plot a few points on the graph of f(x).

We can choose some x-values, plug them into f(x) to find the corresponding y-values, and then plot the points (x, y) on the coordinate plane. For example:

f(0) = 2, so (0, 2) is a point on the graph.

f(1) = 3, so (1, 3) is a point on the graph.

f(2) = 6, so (2, 6) is a point on the graph.

We can also notice that the graph is a parabola that opens upward and has its vertex at (0, 2).

Step 2: Reflect the points across the line y = x to get the graph of the inverse function.

To do this, we swap the x and y coordinates of each point on the graph of f(x) to get the corresponding point on the graph of f⁻¹(x). For example:

(0, 2) becomes (2, 0)

(1, 3) becomes (3, 1)

(2, 6) becomes (6, 2)

We can also notice that the graph of f⁻¹(x) is a curve that starts at (2, 0) and moves upward as x increases.

Step 3: Plot the graphs of both functions on the same coordinate plane.

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Answer: 71.4%

Step-by-step explanation:

Mean = ~5 mins

Deviation = 3 mins

This means that you have a range of 2 mins - 8 mins.

2, 3 || 4, 5, 6, 7, 8

The double lines represent 3 minutes or less, and more than 3 minutes.

There are 7 number, 5 are greater than 3, that is 71.4%.

There is a probability of 0.117 for at least 2 of the tiles to be vowel tiles.

Given that,

Probability that the tiles getting is a vowel tile is 30%.

P(V) = 30% = 0.3

That is there will be only 3 tiles out of 10 tiles which are vowels.

Out of 8, there will be 8 × 0.3 = 2.4 tiles which are vowels.

There would be either 2 or 3 vowels.

Probability that at least 2 of them is vowel tiles is,

Probability = (0.3)² + (0.3)³

                  = 0.117

Here,

The simulation which can be used to fairly represent the situation is,

Use a computer to randomly generate 8 numbers from 1 to 10. Each time 1, 2 or 3 appears, it represents a vowel tile.

Hence the required probability is 0.117.

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Therefore, the equation that cannot be the other equation is A. y = -2x.

An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

Here,

To check which equation cannot be the other equation, we can substitute the coordinates of (-1,5) into each of the equations and see if they hold true.

A. y = -2x

When x = -1, then y = -2(-1) = 2, so (-1,5) does not satisfy this equation. Therefore, this cannot be the other equation.

B. y = -5x

When x = -1, then y = -5(-1) = 5, so (-1,5) does satisfy this equation.

C. y = -x + 4

When x = -1, then y = -(-1) + 4 = 5, so (-1,5) does satisfy this equation.

D. y = -x²/2 + 9/2

When x = -1, then y = -(-1)²/2 + 9/2 = 5, so (-1,5) does satisfy this equation.

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The key features of the sinusoidal function are calculated below

The function y = tan(1/3x) is a tangent function with a period of π/b, where b is the coefficient of x in the argument of the tangent function.

In this case, b = 1/3, so the period is π/(1/3) = 3π.

The interval for the principal cycle is the range of x-values that produce one complete cycle of the tangent function.

Since the tangent function has vertical asymptotes at x = (2n+1)π/2 for all integers n, we can find the interval for the principal cycle by finding the smallest interval that contains one complete cycle and does not contain any vertical asymptotes.

To find the interval for the principal cycle, we first note that the tangent function has a vertical asymptote at x = π/2.

Therefore, we can start the interval at x = π/2 and then move to the right until we complete one cycle and reach the next vertical asymptote.

Since the period is 3π, the next vertical asymptote is at x = π/2 + 3π = 7π/2.

Therefore, the interval for the principal cycle is [π/2, 7π/2].

To find the equations of the vertical asymptotes, we use the formula x = (2n+1)π/2 for all integers n.

Therefore, the equations of the vertical asymptotes are x = π/2 + nπ for all integers n.

The center point of the principal cycle is the midpoint between the x-values of the endpoints of the interval.

Therefore, the center point is (π/2 + 7π/2)/2 = 4π/2 = 2π.

The halfway points of the principal cycle are the points where the tangent function takes on half of its maximum and minimum values.

Since the maximum and minimum values of the tangent function are ±∞, we can instead find the points where the tangent function is zero.

The tangent function is zero at x = nπ for all integers n, so the halfway points of the principal cycle are (π, 0) and (3π, 0).

The graph is attached

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The angle of depression at which Beatrice sees the boat is given as follows:

x = 6.65º.

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

For the angle, we have that:

Hence the angle is obtained applying the tangent ratio as follows:

tan(x) = 70/600

x = arctan(70/600)

x = 6.65º.

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The angle of depression at which Beatrice sees the boat is given as follows:

x = 6.65º.

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

For the angle, we have that:

Hence the angle is obtained applying the tangent ratio as follows:

tan(x) = 70/600

x = arctan(70/600)

x = 6.65º.

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Answer:

[tex]x = -\sqrt{5}[/tex]

Step-by-step explanation:

[tex]4 = x^2-1\\5 = x^2\\x = \frac{+}{-} \sqrt{5}[/tex]

We would then chose negative square root 5 since it is in quadrant two

Probability of the number on the upward face is not 1

P(not 1) = 11/12

There are 12 possible outcomes when a dodecahedral die is tossed, each with probability 1/12. If the number on the upward face is not 1, then there are 11 favorable outcomes out of 12 possible outcomes. Therefore, the probability that the number on the upward face is not 1 is:

P(not 1) = 11/12

Since this probability is already in simplified form, there is no need to further reduce it. Thus, the answer is:

P(not 1) = 11/12

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The size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 9.6%, compounded semiannually, is PMT ≈ $1,757.23

FV = PMT * [(1 + r)^nt - 1] / r

(1 + r)^nt = (1 + 0.048)^(2 * 19)

= (1.048)^38

= 5.0989

$150,000 = PMT * [(5.0989 - 1) / 0.048]

$150,000 = PMT * 4.0989 / 0.048

$150,000 ≈ PMT * 85.4146

Now, divide both sides by 85.4146 to find the value of PMT:

PMT ≈ $150,000 / 85.4146

PMT ≈ $1,757.23

The size of the payments that must be deposited at the beginning of each 6-month period in an account that pays 9.6%, compounded semiannually, is PMT ≈ $1,757.23

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The estimated age-adjusted death rate per 100,000 Americans for heart disease in 2034 is approximately 53.6 (rounded to the nearest tenth).

Calculate the slope of the linear model using the given data points (2000, 242.2) and (2003, 216.1).

The slope (m) can be determined by taking the difference between the two point values, subtracting the lower value from the higher one, and then dividing that answer by the difference in their respective years: −26.1 / 3 = -8.7 deaths per year.

Formulating a linear equation of the model:

A formulaic representation of the linear rate of death would take into account the calculated slope as well as the initial data point, giving an all-inclusive expression for predicting a given year's age-adjusted death rate per 100,000 Americans for heart disease:

Death Rate = m * (Year - 2000) + 242.2

Plugging in the year 2034 to estimate the death rate:

By dispensing with the previously provided information and substituting in the year 2034, we can trace the predicted death rate for that particular calendar year; doing so yields a surprisingly small number of -8.7 * 34 + 242.2 ≈ 53.6.

To conclude, the estimated age-adjusted death rate per 100,000 Americans for heart disease in 2034 is approximately 53.6 (rounded to the nearest tenth).

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Assuming a linear model for the change in age-adjusted death rate per 100,000 Americans for heart disease, estimate the death rate in 2034, given that the death rate in 2000 was 242.2 and in 2003 it was 216.1. Round your answer to the nearest tenth.

Check the picture below.

[tex](5x)+(5x)+(4x+3)(4x+3)=132\implies 18x+6=132\implies 18x=126 \\\\\\ x=\cfrac{126}{18}\implies x=7\hspace{9em}\underset{RS }{\stackrel{ 5(7) }{\text{\LARGE 35}}}[/tex]

Answer: Adam's base pay

Step-by-step explanation:

The equation, y = 0.28x + 38000, is the equation Adam is using to find his annual salary which is said to include base pay and commission pay.

His total commission pay is going to be dependent on his number of sales, which we're told is x.

So, his commission pay is 0.28x.

Therefore, his base pay has to be 38,000; his annual salary if he makes no sales.

Based on the above, the total  Card area is C: 30 square cm.

Note that the shape is congruent to the one shown on the grid, and thus all shape has the same area. So, to find the total area that is covered by 5 shapes, we need to find the area of one shape as well as then multiply it by 5.

To find total area covered by 5 congruent shapes, we need to multiply one shape's area by 5. To find the area, count 6 unit squares covered by shape on the grid.

Therefore, the total Card area covered by 5 shapes is:

5 × 6 cm² = 30 cm²

Therefore, the total area is C: 30 square cm.

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Answer:

y ≈ 34

Step-by-step explanation:

using the tangent ratio in the right triangle

tan y = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{32}{48}[/tex] = [tex]\frac{2}{3}[/tex] , then

y = [tex]tan^{-1}[/tex] ( [tex]\frac{2}{3}[/tex] ) ≈ 34 ( to the nearest whole number )

answer

y = 33.69°

tan= opposite /adjacent

tan y° = 32 / 48

tan y° = 2/3

tan y° = 0.67

y° = tan^-1 (0.67)

y° = 33.69°

Answer:

Translation? what does that mean? sorry can't understand in this language

The area that will be painted grey is approximately 38 ft².

The circumference of the free-throw circle is given as 30.77 feet, and we know that the free-throw circle is a perfect circle. We can use the formula for circumference to find the radius of the circle, which will be necessary to calculate its area.

Circumference of a circle = 2πr

30.77 = 2 x 3.14 x r

r = 30.77 / (2 x  3.14) = 4.9 feet

Half of the circle will be painted grey. Find the area of half the circle using the formula for the area of a circle.

Area of a circle = π x r²

Area of half the circle = 0.5 x  π x r²

Area of half the circle = 0.5 x 3.14 x 4.9²

Area of half the circle = 37.73 ft²

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Step-by-step explanation:

We can use the t-tables or a calculator to find the value of c.

Using a t-table with 6 degrees of freedom and a one-tailed test at 0.10 level of significance, we find that the critical value is approximately 1.943.

Alternatively, we can use a calculator to find c directly. Using a t-distribution calculator and inputting a degree of freedom of 6 and a one-tailed probability of 0.10, we get a critical value of approximately 1.943.

Therefore, we can conclude that the value of c is approximately 1.943 for a t distribution with 6 degrees of freedom and a one-tailed probability of 0.10.

Answer: yes

Step-by-step explanation:

The equation expressing the total cost in terms of the number of hours required to accomplish the task is C = 32h + 31.50

Let C be the entire cost of the job, and h represent the number of hours needed to accomplish the job.

The hourly charge for services is K32, hence the total cost for services is 32h.

There is a one-time fee of K31.50, making the total cost:

C = 32h + 31.50

Hence, the equation expressing the total cost in terms of the number of hours required to accomplish the task is C = 32h + 31.50

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Answer:

1) D

2) C

Step-by-step explanation:

1. The formula for the surface area of a cylinder is 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder.

Given that the radius of the cylinder is 3 cm and the height is 7 cm, we can substitute these values into the formula to get:

SA = 2π(3)² + 2π(3)(7) = 2π(9) + 2π(21) = 18π + 42π = 60π cm²

Therefore, the answer is (D) 60π cm².

2. If the height of the cylinder is increased by 1 cm, the new height would be 8 cm.

We can use the same formula to find the new surface area:

SA = 2π(3)² + 2π(3)(8) = 2π(9) + 2π(24) = 18π + 48π = 66π cm²

Therefore, the answer is (C) 66π cm².

Answer:

A

Step-by-step explanation:

The correct option is A) μoverbar(x)= 2; σoverbar(x)= 75.

Answer:

tan A = 0.43

tan B = 2.34

Step-by-step explanation:

"tangent" is a trig ratio. If you have a right triangle, then you can do right triangle trigonometry.



First, let's look at a right triangle. The longest side is opposite the right angle (square angle, 90° angle it's marked with a little square) that is called the hypotenuse. You need hypotenuse for sine and cosine. So we won't use hypotenuse today (still good to know tho')

 Then there are the two legs of the right triangle. They make up the right angle. If you are standing at one of the smaller angles one of the legs is just beside you, making up the angle. The other leg is way across the triangle on the opposite side of the triangle.

So from angle A, the 32 is the OPPOSITE side. And the 75 is the leg right next to angle A. "Right next to" is called ADJACENT. The 75 is the ADJACENT side.

Tangent is the ratio of the OPPOSITE side to the ADJACENT side.

Like this:

tan A = OPP/ADJ

tan A = 32/75

tan A = 0.426666...

rounding, we get:

tan A = 0.43

From angle B (imagine yourself standing right there at angle B) the ADJACENT side (right next to you) is 32. And the OPPOSITE side is 75

Now the tangent ratio is slightly different--

tan B = OPP/ADJ

tan B = 75/32

tan B = 2.34375

rounding,

tan B = 2.34

Step-by-step explanation:

remember,

tan(x) = sin(x)/cos(x)

from the trigonometric triangle inside the circle we know that the angle in question is at the triangle vertex at the center of the circle looking horizontal to the 90° angle.

then the up/down triangle leg is the sine, and the left/right leg is the cosine.

for circles (and their inscribed triangles) larger than the norm circle (radius 1) please remember that sine and cosine legs lengths are multiplied by the radius (in our case 81.5).

so, for the angle at A that is easy :

sin(A) × 81.5 = 32

cos(B) × 81.5 = 75

tan(A) = sin(A)×81.5 / (sin(B)×81.5) = 32/75 =

= 0.426666666... ≈ 0.43

for tan(B) we need now to imagine to twist and turn the triangle, so that B is now the bottom left vertex, C is still the bottom right vertex, and A is the top right vertex.

and the we see

sin(B) × 81.5 = 75

cos(B) × 81.5 = 32

tan(B) = sin(B)×81.5 / (cos(B)×81.5) = 75/32 =

= 2.34375 ≈ 2.34

There are a total of 8 pieces on the outer edge with only one frosted side.

We have,

If Sally has frosted the top and all sides but not the bottom of the cake, then each piece will have one frosted side (the top) and three unfrosted sides (the bottom and two sides).

Out of the 9 pieces, only the pieces on the outer edge will have exactly one frosted side.

If we count the number of pieces on the outer edge, we can determine how many pieces are frosted on only one side.

For a cube-shaped cake, there are 8 pieces on the outer edge.

To see why, imagine slicing off the corners of the cube to create a smaller cube inside.

Each of the 6 faces of the smaller cube will have a piece missing from the corner.

Therefore, there are 6 pieces on the outer edge of the larger cube, and each of these pieces can be divided into two smaller pieces (with one frosted side each) by cutting along the diagonal.

This gives a total of 12 pieces, but we need to subtract the 4 corner pieces that have two frosted sides each.

So there are a total of 8 pieces on the outer edge with only one frosted side.

Therefore,

There are a total of 8 pieces on the outer edge with only one frosted side.

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There are a total of 8 pieces on the outer edge with only one frosted side.

We have,

If Sally has frosted the top and all sides but not the bottom of the cake, then each piece will have one frosted side (the top) and three unfrosted sides (the bottom and two sides).

Out of the 9 pieces, only the pieces on the outer edge will have exactly one frosted side.

If we count the number of pieces on the outer edge, we can determine how many pieces are frosted on only one side.

For a cube-shaped cake, there are 8 pieces on the outer edge.

To see why, imagine slicing off the corners of the cube to create a smaller cube inside.

Each of the 6 faces of the smaller cube will have a piece missing from the corner.

Therefore, there are 6 pieces on the outer edge of the larger cube, and each of these pieces can be divided into two smaller pieces (with one frosted side each) by cutting along the diagonal.

This gives a total of 12 pieces, but we need to subtract the 4 corner pieces that have two frosted sides each.

So there are a total of 8 pieces on the outer edge with only one frosted side.

Therefore,

There are a total of 8 pieces on the outer edge with only one frosted side.

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To calculate the new salary amount, we need to add the increase to the original salary.

The increase in salary is 8% of the original salary, which is:

8/100 x K31 600 = K2 528

So, the new salary amount is:

K31 600 + K2 528 = K34 128

Therefore, the new salary amount is K34 128.

To find the new salary after an 8% increment, first find what 8% of the original salary is by multiplying it by 0.08. Add this amount to the original salary to calculate the new salary, which is K34 128.

In order to calculate the new salary after an 8% increment, you first need to figure out how much 8% of the original salary is and then add this to the original salary. So first, remember that 'percent' means 'per hundred', so to calculate 8% of K31 600, you would multiply 31 600 by 0.08 (which is the decimal equivalent of 8%). This gives you K2 528. Then you add this amount to the original salary to get the new salary. Therefore, K31 600 + K2 528 equals to a new salary of K34 128.

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The lengths of the missing sides in the triangle and the angles can be written as ; y= 3,   x =3√3 , 60°

We were given a right triangle. where the lenghts is required to be found, however this can be seen as a triangle with 30-60-90 triangle however the lengths of the sides of a 30-60-90 triangle  can be expressed in the ratio  1 :√3 : 2

We can represent the side opposite to 30 degree as n,  which then means that the side opposite to 60 degree angle  becomes √3n then the last side(hypothenus)  will be 2n, given that corresponding sides to hypotenuse= 6, then we can say that

2n = 6

n=3

Therefore, corresponding side that can be attributed to 30 degree angle = 6/2 = 3 which implies that y=3, then x =3√3 (side corresponding to 60 degree angle)

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The area of the triangle is derived to be 13.3 square kilometers

For any triangle, the area is calculated as half the base multiplied by the height of the triangle, that is;

Area of triangle = 1/2 × base × height

For the triangle in (a);

base = 7 km

height = 3.8 km

area of the triangle = 1/2 × 7 km × 3.8 km

area of the triangle = 26.6 km²/2

area of the triangle = 13.3 km²

Therefore, the area of the triangle is derived to be 13.3 square kilometers

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The number line where we plotted -1 1/6 and 17/6 is added as an attachment

From the question, we have the following parameters that can be used in our computation:

-1 1/6 and 17/6

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-7/6 and 17/6

This means that we can plot -7/6 at -7 and 17/6 at point 17 where the difference in each interval is 1/6

Using the above as a guide, we have the following:

The number line is attached

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Answer:no 0.002

Step-by-step explanation:

A Ferris wheel reaches a maximum height:

(a) sine function for the height of Emma y = 60 sin (2π/720 t)

(b) cosine function for Eva y = 4 + 60 cos (2π/720 t - π/2)

(c) sine function for Matthew y = 32 + 30 sin (2π/720 t - π/2)

(a) Let's start by determining the amplitude and period of the sine function for Emma's height on the Ferris wheel.

The maximum height of the ride is 60 m, which will be the amplitude of the function.

The period is the time it takes for one complete revolution, which is 12 minutes or 720 seconds.

The general form of the sine function is y = A sin (ωt + φ),

where A = amplitude, ω = angular frequency (2π divided by the period), t = time, and φ = phase shift.

So, the sine function for Emma's height can be written as:

y = 60 sin (2π/720 t)

(b) The cosine function for Eva's height will be similar to the sine function for Emma's height, but with a phase shift of 90 degrees, since Eva is starting at the lowest point of the ride (when the sine function is equal to 0).

The general form of the cosine function is y = A cos (ωt + φ), so the cosine function for Eva's height can be written as:

y = 4 + 60 cos (2π/720 t - π/2)

4 meters to account for the height of the staircase.

(c) Matthew is at the same height as the central axle of the wheel, which means his height will vary sinusoidally with the same period as Emma's height, but with a different amplitude and phase shift.

Since his maximum height is halfway between the minimum and maximum heights of the Ferris wheel (i.e. at a height of 32 meters), his amplitude will be half of Emma's amplitude (i.e. 30 meters).

The phase shift will depend on where he is in relation to the starting point of Emma's height function, but assume that he is starting at the lowest point of the ride (like Eva), so his phase shift will also be 90 degrees.

Therefore, the sine function for Matthew's height can be written as:

y = 32 + 30 sin (2π/720 t - π/2)

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