Sara sells dolls at her doll store. If she has a doll that she sells for $15.55, what would be the total cost of the doll if the sales tax were 6% of the sale price?

Answer:

(a) The average rate of change of f on the interval [-3,4] is given by:

(1/(4-(-3))) * ∫[a,b] f(x) dx

where a = -3 and b = 4. We can break up the integral into two parts, one over the interval [-3,0] and the other over the interval (0,4]:

(1/7) * [∫[-3,0] √(9-x²) dx + ∫[0,4] (-x+3cos(πx/2)) dx]

For the first integral, we recognize that the integrand is the equation of the top half of a circle with radius 3 centered at the origin. Therefore, we can use the substitution x = 3sin(t), dx = 3cos(t)dt, to get:

∫[-3,0] √(9-x²) dx = ∫[-π/2,0] 9cos²(t) dt = (9/2) * [sin(t)cos(t) + t]_[-π/2,0] = (9π - 81)/4

For the second integral, we can use integration by parts with u = -x and dv = cos(πx/2) dx to get:

∫[0,4] (-x+3cos(πx/2)) dx = [-x²/2 + (6/π)sin(πx/2)]_0^4 = -8

Therefore, the average rate of change of f on the interval [-3,4] is:

(1/7) * [(9π - 81)/4 - 8] = (9π - 145)/28

(b) To find the equation of the tangent line to the graph of f at x = 3, we need to find the slope of the tangent. Since f is not differentiable at x = 0 (due to the cosine term), we need to consider the left and right derivatives separately.

For x < 0, the function is the equation of the top half of a circle with radius 3 centered at the origin, so the slope of the tangent at x = 3 is:

f'(3-) = -√(9-3²)/(3-0) = -√6

For x > 0, we have:

f'(x) = -1 - (3π/4)sin(πx/2)

So the slope of the tangent at x = 3 is:

f'(3+) = -1 - (3π/4)sin(3π/2) = -1 + (3π/4)

The equation of the tangent line is therefore:

y - f(3) = f'(3)(x-3)

y + √(9-3²) = (-√6)(x-3) (for x < 0)

y - 6 + 3cos(π/2) = [(-1 + (3π/4))(x-3)] (for x > 0)

(c) The average value of f on the interval [-3,4] is given by:

(1/(4-(-3))) * ∫[-3,4] f(x) dx

Using the same breakdown of the integral as in part (a), we have:

(1/7) * [∫[-3,0] √(9-x²) dx + ∫[0,4] (-x+3cos(πx/2)) dx]

The first integral was evaluated in part (a

Step-by-step explanation:

Answer:

85.5

Step-by-step explanation:

85 • 9 is 765

If you add 90 to 765 and then divide the sum by 10, you get 85.5.

According to the description, element a21 will represent B. 100.

According to the question, the rows represent the type of music while the columns represent the weeks. Now the element that we have is a21. The 2 in the element stands for the rows which is the type of music and this is R and B.

The column is the weeks and since we have the column as 1, we will look at week 1 and the second bar in that week which is R and B. So, the correct description is option B.

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The probability that the next puck Dave grabs will be green is 1/6 or approximately 0.167.

To determine the probability of Dave grabbing a green puck on the next grab, we need to calculate the probability based on the given data.

Given:

Number of red pucks = 4

Number of orange pucks = 3

Number of yellow pucks = 1

Number of pink pucks = 2

Number of green pucks = 2

Total number of pucks = 4 + 3 + 1 + 2 + 2 = 12 pucks

The probability of grabbing a green puck can be calculated as:

Probability = Number of green pucks / Total number of pucks

Probability = 2 green pucks / 12 pucks

Probability = 1/6

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The two numbers are -15 and -7.

We have,

To solve this problem, we need to find two numbers that multiply to 105 and add up to -22.

We can start by listing the factors of 105: 1, 3, 5, 7, 15, 21, 35, and 105.

Then, we can try adding pairs of factors to see if we get -22.

We have the system of equations:

xy = 105

x + y = -22

We can solve for one variable in terms of the other using the second equation:

y = -22 - x

Then, we can substitute this into the first equation:

x(-22 - x) = 105

Expanding and rearranging, we get:

x² + 22x + 105 = 0

Now, we can use the quadratic formula to solve for x:

x = (-22 ± √(22² - 4(1)(105))) / 2

x = (-22 ± 4) / 2

x = -15 or x = -7

Thus,

The two numbers are -15 and -7.

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The length of side AC is approximately 2.87 units when rounded to the nearest hundredth.

In a right triangle ABC, where angle ACB is a right angle and angle ABC is 35 degrees, we are given that side AB has a length of 5 units. We need to find the length of side AC.

To find the length of side AC, we can use trigonometric ratios. In this case, we can use the sine function.

The sine of angle ABC is defined as the ratio of the length of the side opposite the angle (AC) to the length of the hypotenuse (AB).

sin(35°) = AC / 5

To find the length of AC, we can rearrange the equation:

AC = 5 * sin(35°)

Using a calculator to find the sine of 35 degrees, we get:

AC ≈ 5 * 0.5736 ≈ 2.868

Therefore, the length of side AC is approximately 2.87 units when rounded to the nearest hundredth.

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The minor arc is QS and the measure of the arc is 98°

An arc is a smooth curve joining two endpoints. It can also be defined as the portion of a circumference of a circle. The circumference of a circle will be divided into minor arc and major arc.

The major arc is the arc that bounds the big sector and the minor arc is the arc that bounds the small sector.

A sector is an area bounded radii and an arc

Therefore the minor arc is arc SQ and the measure of the arc is also 98°. This is because the angle substended by the arc is equal to the measure of the arc.

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Answer: y = 2(x + 2)² - 5

Step-by-step explanation:

          We are going to use the completing the square method to transform this quadratic equation from standard form to vertex form.

Given:

     y = 2x² + 8x + 3

Factor the 2 out of the first two terms:

     y = 2(x² + 4x) + 3

Add and subtract [tex]\frac{b}{2} ^2[/tex]:

     y = 2(x² + 4x + 4 - 4) + 3

Distribute the 2 into -4 and combine with the 3:

     y = 2(x² + 4x + 4) - 5

Factor (x² + 4x + 4):

     y = 2(x + 2)² - 5

Answer:

$0.96

Step-by-step explanation:

ok so 30.72 divided by 2 is 15.36. So 1 gallon is $15.36.

there are 16 cups in a gallon.

So there are 32 cups.

$15.36 dollars in gallons converted to that of 16 cups, or $30.72 dollars in gallons to 32 cups..

30.72 divided by 32 is 0.96. 15.36 divided by 16 is 0.96.

So, $0.96 is the price per cup

To calculate the inductance L in the circuit, we can use the formula for the maximum current in an inductive circuit:

Imax = ΔVmax / (ωL)

where Imax is the maximum current, ΔVmax is the maximum voltage, ω is the angular frequency, and L is the inductance.

The inductance L is 0.15625 H (or 156.25 mH).

Given that Imax = 8.00 A, ΔVmax = 100 V, and ω = 80.0 Hz, we can rearrange the formula to solve for L:

L = ΔVmax / (Imax * ω)

Substituting the given values:

L = 100 V / (8.00 A * 80.0 Hz)

L = 0.15625 H (or 156.25 mH)

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The tangential component is 4t/(3 + 12t^2)(1 - t^2)i + 8t^2/(3 + 12t^2)j and the normal component is (-12t)/(3 + 12t^2)(1 - t^2)i + [24(3 + 4t^2)]/(3 + 12t^2)j.

We can start by finding the velocity and acceleration vectors:

r(t) = 2(3t - t^3)i + 6t^2j

v(t) = dr/dt = 6(1 - t^2)i + 12tj

a(t) = dv/dt = -12ti + 24j

To find the tangential and normal components of the acceleration vector, we need to project it onto the velocity vector. Let's call the tangential component aT and the normal component aN. Then:

aT = projv a = (a ⋅ v/|v|^2)v = [(0)(6(1 - t^2)) + (24)(12t)]/[(6(1 - t^2))^2 + (12t)^2](6(1 - t^2)i + 12tj)

aT = (24t)/(36 + 144t^2)(6(1 - t^2)i + 12tj)

aT = 4t/(3 + 12t^2)(1 - t^2)i + 8t^2/(3 + 12t^2)j

To find the normal component, we subtract the tangential component from the acceleration vector:

aN = a - aT

aN = (-12t)i + 24j - [4t/(3 + 12t^2)(6(1 - t^2)i + 12tj)]

aN = (-12t)/(3 + 12t^2)(1 - t^2)i + [24(3 + 4t^2)]/(3 + 12t^2)j

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The number of tickets were sold are; 150 for adult tickets and 250 for child tickets

There are 400 seats in the auditorium which means that there are 400 tickets to be sold and the total ticket sales were 2600 dollars.

Since there were adult tickets for 9 each and child tickets for 5 dollars each.

9x + 5y = 2600

x + y = 400

Solving;

9x + 5(400-x) = 2600

9x + 2000 - 5x = 2600

4x = 600

x = 150

Then y = 250

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Step-by-step explanation:

Use the equation, A = p(1+ r/n)^Tn

n = 12 because it's monthly

T = 20

P = 25,000

r = .01

plug it all in and that question A i think

The value of b in Rhombus UVWX is 86 degrees.

In a rhombus, opposite angles are congruent, so we have:

Angle W = Angle U = 90 degrees (since UVWX is a rhombus)

Angle V = Angle X (opposite angles are congruent)

Using the fact that the sum of the angles in a quadrilateral is 360 degrees, we can write an equation in terms of b:

Angle U + Angle V + Angle W + Angle X = 360 degrees

Substituting the known values, we get:

90 + Angle V + 90 + Angle X = 360

Simplifying, we get:

Angle V + Angle X = 180

Substituting Angle V = 2b - 62 and Angle X = b - 16, we get:

2b - 62 + b - 16 = 180

Simplifying, we get:

3b - 78 = 180

Adding 78 to both sides, we get:

3b = 258

Dividing by 3, we get:

b = 86

Therefore, the value of b in rhombus UVWX is 86 degrees.

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To estimate the required sample size for a pharmaceutical company developing a new drug, we need to consider the proportion of effectiveness, desired margin of error, and confidence interval. Therefore, approximately 1846 patients should be included in the sample to achieve the desired margin of error for a 98% confidence interval.

To estimate the proportion better, the pharmaceutical company needs to increase their sample size until the margin of error is less than 0.005 for a 98% confidence interval. The margin of error is the amount of error that is allowed in a study and is determined by the sample size. The larger the sample size, the smaller the margin of error.
To calculate the sample size, we can use a formula that includes the level of confidence, margin of error, and the estimated proportion. Since the drug was found to be 80% effective, we can use this as our estimated proportion.
The formula to calculate the sample size is:
n = (Z^2 * p * q) / E^2
where n is the sample size, Z is the z-score corresponding to the desired level of confidence (2.33 for 98% confidence interval), p is the estimated proportion (0.8), q is 1-p (0.2), and E is the desired margin of error (0.005).
Plugging in the values, we get:
n = (2.33^2 * 0.8 * 0.2) / 0.005^2
n = 23474.4
Rounding up to the nearest whole number, the pharmaceutical company should sample at least 23475 patients to achieve a margin of error less than 0.005 for a 98% confidence interval.
To estimate the required sample size for a pharmaceutical company developing a new drug, we need to consider the proportion of effectiveness, desired margin of error, and confidence interval. In this case, the drug is 80% effective, and the company wants a margin of error less than 0.005 for a 98% confidence interval.
To calculate the sample size, we use the formula for sample size estimation in proportion:
n = (Z^2 * p * (1-p)) / E^2
where n is the sample size, Z is the Z-score corresponding to the desired confidence interval, p is the proportion of effectiveness (0.8 in this case), and E is the desired margin of error (0.005).
For a 98% confidence interval, the Z-score is approximately 2.33. Plugging the values into the formula:
n = (2.33^2 * 0.8 * (1-0.8)) / 0.005^2
n ≈ 1846
Therefore, approximately 1846 patients should be included in the sample to achieve the desired margin of error for a 98% confidence interval.

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If the number of research papers in Dr. Carey's area of expertise is increasing by 3% every year and 34,176 research papers were published this year, then approximately 35,168 research papers will be published two years from now.

To calculate the number of research papers that will be published two years from now, we need to apply the 3% annual increase for two consecutive years to the initial count of 34,176 papers.

First, we calculate the increase for the first year:

Increase = 34,176 * 3% = 34,176 * 0.03 = 1,025.28 (approximately)

Next, we add this increase to the initial count to get the count after one year:

Count after one year = 34,176 + 1,025.28 = 35,201.28 (approximately)

Now, we calculate the increase for the second year based on the count after one year:

Increase = 35,201.28 * 3% = 35,201.28 * 0.03 = 1,056.04 (approximately)

Finally, we add this increase to the count after one year to get the count after two years:

Count after two years = 35,201.28 + 1,056.04 = 36,257.32 (approximately)

Therefore, approximately 36,257 research papers will be published two years from now.

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Answer: The circumference of a circle is given by the formula:

C = πd

where d is the diameter of the circle. In this case, the diameter of the wheels is 28 inches, so the circumference of each wheel is:

C = π(28) = 28π inches

To find how many revolutions the wheels need to make to travel 200 feet, we need to convert 200 feet to inches, and then divide by the circumference of each wheel. There are 12 inches in 1 foot, so 200 feet is equal to:

200 feet × 12 inches/foot = 2400 inches

Dividing 2400 inches by the circumference of each wheel, we get:

2400 inches ÷ (28π inches/revolution) ≈ 85.3 revolutions

Therefore, the car's wheels need to make approximately 85.3 full revolutions to travel 200 feet. Since the question asks for the number of full revolutions, we can round down to the nearest whole number to get:

Answer: D. 88 revolutions.

The surface area of the cylinder is 32π units²

What is surface area of cylinder?

A cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance. The base of a cylinder is circular and it's volume is given by ; V = πr²h

The surface area of a cylinder is expressed as;

SA = 2πr( r+h)

where r is the radius and h is the height.

radius = 2 units

height = 6 units

SA = 2×2 π( 2+6)

SA = 4π × 8

SA = 32π units²

Therefore the surface area of the cylinder in term of pi is 32π units².

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Answer:

To find the inverse function of f(x) = 4x + 12, we follow these steps:Replace f(x) with y: y = 4x + 12.Swap the variables x and y: x = 4y + 12.Solve for y in terms of x: y = (x - 12) / 4.Therefore, the inverse function of f(x) is f^-1(x) = (x - 12) / 4.Now, we can verify that f(f^-1(x)) = x and f^-1(f(x)) = x as follows:f(f^-1(x)) = f((x - 12) / 4) [substitute f^-1(x) into f(x)]

= 4((x - 12) / 4) + 12 [substitute (x - 12) / 4 into 4x + 12]

= x [simplify]Therefore, f(f^-1(x)) = x.f^-1(f(x)) = ((4x + 12) - 12) / 4 [substitute f(x) into f^-1(x)]

= x / 4 [simplify]Therefore, f^-1(f(x)) = x/4.Since f(f^-1(x)) = x and f^-1(f(x)) = x/4, we have verified that the inverse function of f(x) satisfies the conditions of an inverse function.

Step-by-step explanation:

The parallel vectors to CD are:

Here we have the vector:

CD = 4a - 3b

Where a and b are vectors.

A vector V will be parallel to CD if we can find a scalar number k such that we can write the vector as:

V = k*CD

The second option:

V = 20a - 15b

We can rewrite this as:

V = 20a - 15b = 5*(4a - 3b)   this vector is parallel.

The third one is:

V'  =10a - 20b + 2a + 11b

V' = 12a - 9b = 3*(4a - 3b)  this vector is parallel.

V'' = (-8a + 6b) = -2*(4a - 3b)  this vector is parallel.

V''' = a - (3/4)b = (1/4)*(4a - 3b)  this vector is parallel.

These are all the parallel vectors.

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The data in the table represent a direct variation

The equation is y = 5x

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

x  1 3 4 7

y 5 15 20 35

In the above table of values, we can see that

As the x values, the y values increase

This means that the table of values represent a direct variation

The equation is a direct variation

So, we have

y = kx

The value of k is calculated as

k = 5/1

So, we have

k = 5

This means that

y = 5x

Hence, the equation is y = 5x

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Answer:

Step-by-step explanation:

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and my latest song The Light

999 Forever

Mean for start of school year is 6.5; Mean for end of school year is 7.2.

Median for start of school year is 6.5; Median for end of school year is 7.

To find the means, list out each data value given for each dot plot and calculated the mean.

Mean for start of school year:

We have, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9

Mean = ( 4 + 5 + 5 + 6 + 6 + 6 + 6 + 7 + 7 + 7 + 7 + 8 + 8 + 9)/14

= 91/14

Mean ≈ 6.5

Mean for end of school year:

We have, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9

Mean = ( 5 + 5 + 6 + 6 + 7 + 7 + 7 + 7 + 8 + 8 + 8 + 9 + 9 + 9)/14

= 101/14

Mean ≈ 7.2

Median represents the middle data value in a data set, therefore:

Median for start of school year = ( 6 + 7)/2 = 6.5

Median for end of school year = ( 7 + 7)/2 = 7

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There is no meaningful difference in the weights of the two types of rings

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

Platinum = 8.21 grams

Gold = 8.61 grams

The difference in these weights is

Difference = 8.61 grams - 8.21 grams

Evaluate the difference

Difference = 0.4 grams

This value approximates to 0

This means that there is no meaningful difference in the weights

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The requried measures of DEF and x are 82° and 90° respectively.

In a quadrilateral, the sum of the opposite angle is 180°,
So,
98 + y = 180
y = 180 - 98
y =  82°

Similarly,
x + x = 180
2x = 90
x = 90°

Thus, the requried measures of DEF and x are 82° and 90° respectively.

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Using the tangent of the angle from trigonometric ratio, the value of x is 4.0

In trigonometry, there are six trigonometric ratios, namely, sine, cosine, tangent, secant, cosecant, and cotangent. These ratios are written as sin, cos, tan, sec, cosec(or csc), and cot in short.

sin θ = opposite / hypothenuse

cos θ = adjacent / hypothenuse

tan θ = opposite / adjacent

To solve this problem, we can use the tangent of the angle.

tan θ = opposite / adjacent

tan 20 = x / 11

Cross multiply both sides;

x = 11 tan 20

x = 4.0

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The distance between two points a and b on the real line is given by the absolute difference between the two points, which is calculated as the positive difference between the values of a and b regardless of their order.

The distance function, denoted as d(a, b), is a metric that satisfies the properties of non-negativity, symmetry, and the triangle inequality. It is used to quantify the distance between two points in one-dimensional space, and is an important concept in geometry, analysis, and other fields of mathematics. The distance formula can be extended to higher dimensions and is used in various applications such as optimization, clustering, and machine learning.

The distance between two points a and b on the real line is given by the absolute difference between the two points: d(a, b) = |a - b|

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The mean weight of a randomly chosen vehicle can be calculated by taking the average of the minimum and maximum weights:

Mean = (2,180 + 4,449) / 2 = 3,314.5 pounds

The standard deviation of a uniformly distributed random variable can be calculated using the following formula:

Standard Deviation = (Max - Min) / √12

Standard Deviation = (4,449 - 2,180) / √12 ≈ 652.48 pounds

To find the probability that a vehicle will weigh less than 2,389 pounds, we need to calculate the proportion of the total range that falls below 2,389 pounds:

Probability = (2,389 - 2,180) / (4,449 - 2,180) ≈ 0.317

To find the probability that a vehicle will weigh more than 3,672 pounds, we need to calculate the proportion of the total range that exceeds 3,672 pounds:

Probability = (4,449 - 3,672) / (4,449 - 2,180) ≈ 0.361

To find the probability that a vehicle will weigh between 2,389 and 3,672 pounds, we need to calculate the proportion of the total range that falls within this interval:

Probability = (3,672 - 2,389) / (4,449 - 2,180) ≈ 0.322

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