Suppose there are 5 people and 4 waffle. What is each person's share of 4 waffles?

The derivative of y = x^(sin x) with respect to x is:

dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x)).

To find the derivative of y = x^(sin x) with respect to x using logarithmic differentiation, follow these steps:
1. Take the natural logarithm of both sides of the equation:

ln(y) = ln(x^(sin x))
2. Use the properties of logarithms to simplify:

ln(y) = sin x * ln(x)
3. Differentiate both sides with respect to x, using the chain rule and product rule:

(1/y) * dy/dx = cos x * ln(x) + sin x * (1/x)
4. Multiply both sides by y to solve for dy/dx:

dy/dx = y * (cos x * ln(x) + sin x * (1/x))
5. Substitute the original expression for y (y = x^(sin x)) back into the equation:

dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x))
So the derivative of y = x^(sin x) with respect to x is:

dy/dx = x^(sin x) * (cos x * ln(x) + sin x * (1/x)).

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Answer:

x = 16

Step-by-step explanation:

Multiply the entire first equation by -5 and the entire second equation by 2.

You then get:

15x + 20y = 200

2x - 20y = 72

Add the two equations and you get:

17x = 272

Divide 17 from both sides and you get the answer you need:

x = 16

The cost of each item, obtained from the equation for the sum of the costs of the item are;

A brush costs $3.5

A sketchbook costs $10.5

A paint set costs $21

An equation is a mathematical statement that expresses equivalence between two expression joined by an '=' sign.

The amount Camille brought to the art supply = $39.50

The cost of the brush = (1/3) × The cost of the sketchbook

Cost of the sketchbook = (1/2) × Cost of the paint set

Amunt Camille had left over = $4.50

The cost of the items Camille bought = $39.50 - $4.5 = $35

Let x represent the cost of the brush, let y represent the cost of the sketchbook and let z represent the cost of the paint set

Therefore, we get the following equation; x + y + z = 35

x = (1/3)·y

y = (1/2)·z

Which indicates;

x = (1/3) × (1/2)·z = (1/6)·z

From which we get; (1/6)·z + (1/2)·z + z = 35

(5/3)·z = 3

z = 35 × 3/5 = 21

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(a) The critical numbers happen when x = 3 or x = -1/2

(b) f is decreasing on (-∞, -1/2), increasing on (-1/2, 3), and increasing on (3, ∞).

(c) f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.

(d) f is concave downward on (-∞, 1/4) and concave upward on (1/4, ∞).

(e) The inflection point of f is at x = 1/4.

(a) To find the critical numbers of f, we need to find the values of x where the derivative of f equals zero or does not exist.

f'(x) = 12x² - 6x - 18 = 6(2x² - x - 3) = 6(x - 3)(2x + 1)

Setting f'(x) equal to zero, we get:

6(x - 3)(2x + 1) = 0

x = 3 or x = -1/2

These are the critical numbers of f.

(b) To find the intervals where f is increasing and decreasing, we need to examine the sign of the derivative f'(x) in the intervals determined by the critical numbers.

When x < -1/2, f'(x) < 0, so f is decreasing on the interval (-∞, -1/2).

When -1/2 < x < 3, f'(x) > 0, so f is increasing on the interval (-1/2, 3).

When x > 3, f'(x) > 0, so f is increasing on the interval (3, ∞).

(c) To find the local minimum and maximum values of f, we need to examine the critical numbers and the end points of the intervals.

f(3) = 4(3)³ - 3(3)² - 18(3) + 5 = -22

f(-1/2) = 4(-1/2)³ - 3(-1/2)² - 18(-1/2) + 5 = 25.5

Thus, f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.

(d) To find the intervals where f is concave upward and concave downward, we need to examine the sign of the second derivative f''(x).

f''(x) = 24x - 6 = 6(4x - 1)

When x < 1/4, f''(x) < 0, so f is concave downward on the interval (-∞, 1/4).

1/4 < x, f''(x) > 0, so f is concave upward on the interval (1/4, ∞).

(e) To find the inflection points of f, we need to examine the points where the concavity changes.

The concavity changes at x = 1/4, which is the only inflection point o

The equation of the directrix is X = 6 (Option C).

To determine the equation of the directrix of the polar equation r = 26.4/(4 + 4.4cos(theta)), we need to find the constant value of either x or y. This equation is in the form r = ed/(1 + ecos(theta)), where e is the eccentricity, and d is the distance from the pole to the directrix.

In our case, 26.4 = ed and 4.4 = e. To find the value of d, we can divide 26.4 by 4.4:

d = 26.4 / 4.4 = 6

Since the directrix is a vertical line, it has the form x = constant. In this case, the constant is 6.

So, the equation of the directrix is X = 6 (Option C).

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To estimate a population mean with 68.26% confidence that the sample mean will not differ from the population mean by more than 4 units, a sample size of 7 is needed. So, the correct answer is B).

The formula to calculate the sample size needed to estimate the population mean with a specified margin of error, assuming the population standard deviation is known, is

n = ((z-score * σ) / E)²

where

n = sample size

z-score = the z-score corresponding to the desired confidence level (in this case, the 68.26% confidence level corresponds to a z-score of 1)

σ = population standard deviation

E = the desired margin of error

Substituting the given values, we get

n = ((1 * √(99)) / 4)²

n = 6.1875

Since we need to have a whole number for the sample size, we must round up to the nearest integer. Therefore, the necessary sample size is 7.

So, the answer is B) 7.

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Answer:

Step-by-step explanation:

DE/EC.

The expression that shows the total number of pages in Peyton's album is  6 1/2 + f.

We are given that;

Number of pages= 6 1/2

Now,

To write an expression that shows the total number of pages in Peyton’s album, you need to add the number of pages of family photos and the number of pages of friends photos. The expression is:

6 1/2 + f

To evaluate the expression if there are 3 1/2 pages of photos of friends, you need to substitute f with 3 1/2 and then add the fractions. The answer is

6 1/2 + 3 1/2 = 10

Therefore, by the expression the answer will be 6 1/2 + f.

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The containment area is shrinking at a rate of 280π m²/min when the diameter is 80m and the radius is decreasing at a constant rate of 7m/min.

Let's begin by first finding the radius of the containment area when its diameter is 80m.

The diameter of the containment area is 80m, so its radius is half of that:

[tex]r = 80m / 2 = 40m[/tex]

Now, we need to find the rate at which the containment area is shrinking when the radius is decreasing at a constant rate of 7m/min.

We can use the chain rule of differentiation to find this rate:

[tex]dA/dt = dA/dr * dr/dt[/tex]

where A is the area of the containment, t is time, r is the radius of the containment, and dA/dt and dr/dt are the rates of change of A and r with respect to time, respectively.

We know that dr/dt = -7 m/min (negative because the radius is decreasing), and we can find dA/dr by differentiating the formula for the area of a circle with respect to r:

A = π[tex]r^2[/tex]

[tex]dA/dr = 2πr[/tex]

So, when r = 40m, we have:

[tex]dA/dt = dA/dr * dr/dt[/tex]

= (2πr) * (-7)

= -280π [tex]m^2[/tex]/min

Therefore, the containment area is shrinking at a rate of 280π m^2/min when the radius is decreasing at a constant rate of 7m/min and the diameter of the containment area is 80m.

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A reasonable scale factor that could be applied to Shape 2 to create Shape 1 is 0.5.

When we talk about scaling a shape, we mean changing the size of the shape while maintaining its overall proportions.

This can be done by multiplying all of the dimensions of the shape by the scale factor.

For example, if we wanted to make a shape twice as big, we would multiply all of its dimensions (length, width, and height) by 2. If we wanted to make it half as big, we would multiply all of its dimensions by 0.5.

Looking at the two shapes, we can see that Shape 1 is half the size of Shape 2 in all dimensions.

For example, the height of Shape 1 is half the height of Shape 2, the width of Shape 1 is half the width of Shape 2, and the length of Shape 1 is half the length of Shape 2.

Therefore, to transform Shape 2 into Shape 1, we need to multiply all of its dimensions by 0.5. This will result in a scaled copy of Shape 2 that is identical in shape to Shape 1.

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The derivative of y with respect to x when y = e⁻¹⁷ˣ  is dy/dx = -17eˣ

Find the derivative?

To find the derivative of y with respect to x when y = e⁻¹⁷ˣ, we'll use these terms: "derivative", "respect", and "with".

Identify the function. In this case, y = e⁻¹⁷ˣ

Find the derivative (dy/dx) of the function with respect to x. To do this, we'll apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

The outer function is e^(u) (where u is the inner function), and its derivative with respect to u is e^(u). The inner function is -17x, and its derivative with respect to x is -17.

Apply the chain rule. The derivative of y with respect to x (dy/dx) is the product of the derivative of the outer function and the derivative of the inner function: (e^(u)) * (-17).

Substitute u with the inner function (-17x). So, dy/dx = (e⁻¹⁷ˣ * (-17).

The derivative of y with respect to x when y = e⁻¹⁷ˣ  is dy/dx = -17eˣ

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If the perimeter of a semicircle is 35. 98 millimeters, 7 mm is the semicircle's radius.

A semi-circle refers to half of the circle. The circle is cut along the diameter to form a semi-circle.

A diameter is a line segment that passes through the center of the circle and touches the boundary of the circle from both ends.

The perimeter of the semi-circle is the sum of the length of the diameter and the circumference of the semi-circle.

P = 2r + πr

where P is the perimeter

r is the radius

P = 35.98 mm

35.96 = 2r + 3.14r

35.96 = 5.14r

r = 7 mm.

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Integrate the absolute value of the velocity function over each subinterval, and sum up the results to find the distance traveled.

Remember, displacement is the net change in position, while distance traveled is the total length of the path the particle moves along.

Hi! To find the displacement and distance traveled during the time interval [-2, 5], we need to integrate the velocity function v(t) = -t^2 + 5t - 4 over the given interval.

First, let's find the antiderivative of v(t) which gives us the position function s(t):
s(t) = ∫(-t^2 + 5t - 4) dt = (-1/3)t^3 + (5/2)t^2 - 4t + C

For displacement, we simply need to find the difference in the position function at the endpoints of the interval:
displacement = s(5) - s(-2)

For distance traveled, we need to consider both the positive and negative parts of the velocity function. Find the time when v(t) = 0 to determine when the particle changes direction:
-t^2 + 5t - 4 = 0

Solve this quadratic equation for t. Next, divide the interval [-2, 5] into subintervals based on the values of t where the particle changes direction. Integrate the absolute value of the velocity function over each subinterval, and sum up the results to find the distance traveled.

Remember, displacement is the net change in position, while distance traveled is the total length of the path the particle moves along.

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Answer: around 2.6 cm because I rounded to the tenth.

Step-by-step explanation:

r^2=170/8.1×3.14

r^2=170/25.434

r^2≈6.68

Next square root both sides so r^2 becomes r and 6.68 square rooted is about 2.6 cm is the radius.

R≈2.6cm

Answer:

Step-by-step explanation:

The order of division affects the result; 3 ÷ 1/5 is 15 and 1/5 ÷ 3 is 1/15.

To find the answer, we can calculate the quotient of three divided by one-fifth, which is:

3 ÷ (1/5) = 15

And the quotient of one-fifth divided by three is:

(1/5) ÷ 3 = 1/15

These two quotients are different because the order of division changes the result. In the first case, we divide 3 by a smaller number (one-fifth), which results in a larger quotient (15). In the second case, we divide a smaller number (one-fifth) by a larger number (three), which results in a smaller quotient (1/15).

To give a story describing each situation:

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You need to add 2125 milliliters of water to fill the square aquarium completely.

We need to find the volume of the square aquarium and then determine the additional water needed to fill it completely. Here are the steps:

1. Convert the given length to meters: 15 cm = 0.15 m
2. Calculate the volume of the square aquarium: Volume = length × width × height. Since it's a square aquarium, all sides are equal, so Volume = 0.15 m × 0.15 m × 0.15 m = 0.003375 cubic meters.
3. Convert the volume to milliliters: 0.003375 cubic meters × 1,000,000 mL/cubic meter = 3375 mL.
4. Calculate the additional water needed: Total volume - Current volume = 3375 mL - 1250 mL = 2125 mL.

You need to add 2125 milliliters of water to fill the square aquarium completely.

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The number of solutions which this system of linear equations represented in the graph have is: C. Infinitely many solutions.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-1 + 2)/(2 - 0)

Slope (m) = 1/2

At data point (0, -2) and a slope of 1/2, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y + 2 = 1/2(x - 0)  

y = 1/2(x) - 2

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Answer:

Set your calculator to degree mode.

Using the Law of Sines:

7/sin(40°) = x/sin(81°)

x = 7sin(81°)/sin(40°) = 10.8

Answer:

10.8=x

Step-by-step explanation:

Using the Law of Sines, we can put together the fact that

[tex]\frac{sin A}{a} =\frac{sinB}{b}[/tex]

Substitute our given values from the triangle:

[tex]\frac{sin 81}{x} =\frac{sin40}{7}[/tex]

Turn the sines into a decimal:

[tex]\frac{0.9876}{x} =\frac{0.6427}{7}\\[/tex]

cross multiply using butterfly method

0.988·7=0.643x

solve for x

6.916=0.643x

divide both sides by 0.643

10.8=x (round to nearest tenth)

Hope this helps! :)

To calculate the percentage of one's monthly budget that transportation costs account for, we need to know the total amount of money spent on transportation and the total monthly budget.

Let's say, for example, that John spends $500 per month on transportation and his monthly budget is $2,000.

To calculate the percentage, we would divide the amount spent on transportation by the total monthly budget and then multiply by 100 to get the percentage. So, in this case, the calculation would be:

[tex]($500 / $2,000) x 100 = 25%[/tex]

Therefore, John's transportation costs account for 25% of his monthly budget. This is a significant portion of his budget, and if he needs to save money, he may want to consider alternative modes of transportation such as carpooling,

public transportation, or biking. It's always important to keep track of expenses and prioritize spending in order to maintain a healthy financial situation.

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If circle has diameter lm and chord pq, lm = 20 cm, and pq = 16 cm, the length of RM is 10√2 centimeters.

In a circle, a diameter is a chord that passes through the center of the circle. Therefore, the point where the diameter and the chord intersect, in this case, point R, bisects the chord.

Since LM is a diameter, its length is twice the radius of the circle, which means LM = 2r. Thus, we can find the radius of the circle by dividing the diameter by 2: r = LM/2 = 20/2 = 10 cm.

Since point R bisects the chord PQ, RP = RQ = 8 cm (half of PQ). Thus, we need to find the length of RM. To do that, we need to use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, we have a right triangle RLM with RM as the hypotenuse, so we can use the Pythagorean theorem as follows:

RM² = RL² + LM²

RM² = (10)² + (10)²

RM² = 200

RM = √200 = 10√2 cm

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Complete question is:

Consider circle o with diameter lm and chord pq.

if lm = 20 cm, and pq = 16 cm, what is the length of rm, in centimeters?

The equation of the line that best fits the given data is y = (5/6)x + 1/3

The equation of the line that best fits the given data can be found by using the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Using the points (1, 1.5) and (4, 4), we get:

m = (4 - 1.5) / (4 - 1) = 2.5 / 3 = 5/6

Now we can use one of the given points to find the y-intercept. Let's use the point (2, 2):

y = mx + b
2 = (5/6)(2) + b
2 = 5/3 + b
b = 2 - 5/3
b = 1/3

Therefore, the equation of the line that best fits the given data is:

y = (5/6)x + 1/3

The best answer is C. Y = x + 1.

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The antiderivative of the given integral is (X^2/2) + 50X + C, where C is the constant of integration.

This is obtained by integrating the given polynomial directly without the need for trigonometric substitution.First, let's rewrite the integral: ∫(X + 34 + 16) dx. Since the integrand is a polynomial, we don't need trigonometric substitution. Instead, we can find the antiderivative directly:

∫(X + 34 + 16) dx = ∫(X + 50) dx.

Now, find the antiderivative:

(X^2/2) + 50X + C, where C is the constant of integration.

So, the antiderivative of ∫(X + 34 + 16) dx is (X^2/2) + 50X + C.

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If you have access to the information on the back of Ken's bank statement, you can calculate his revised statement balance by adding any credits and subtracting any debits from the previous statement balance.

Reconciling a bank account involves comparing the transactions in your own financial records with those listed on your bank statement. The goal is to ensure that the account balance in your financial records matches the balance reported by the bank.

To reconcile a bank account, you typically start with the ending balance on the previous bank statement, which becomes the beginning balance on the current statement. You then compare this balance with the transactions listed on the current statement, adding any credits (such as deposits or interest payments) and subtracting any debits (such as withdrawals or fees).

The resulting balance should match the ending balance listed on the current bank statement. If it does not match, then there may be errors or discrepancies in the account that need to be investigated. This can involve reviewing bank records, receipts, and other financial documents to identify any errors or missing transactions.

Reconciling your bank account on a regular basis is important for ensuring the accuracy of your financial records and identifying any issues or errors in a timely manner. It can also help you identify areas where you may be overspending or where you can save money by reducing fees or optimizing your financial habits.

Whether or not Ken's account reconciles depends on whether the calculated revised statement balance matches the bank's reported statement balance. If they match, then the account is reconciled. If they do not match, then there may be discrepancies in the account that need to be investigated and resolved.

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Probability of a student participating in football: 0.4 or 40%

Probability of a student participating in tennis: 0.2 or 20%

Assuming that the table lists the number of students who participate in each sport out of a total of 150 students, we can find the probability of a student participating in each sport by dividing the number of students who participate in each sport by the total number of students:

Probability of a student participating in football:

Number of students who participate in football / Total number of students = P(Football)

Probability of a student participating in tennis:

Number of students who participate in tennis / Total number of students = P(Tennis)

For example, if the table shows that 60 students participate in football and 30 students participate in tennis out of a total of 150 students, then the probabilities would be:

Probability of a student participating in football:

60/150 = 0.4 or 40%

Probability of a student participating in tennis:

30/150 = 0.2 or 20%

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The value of the test statistic z is approximately -2.12. The Option D is correct.

To test the hypothesis that the mean wrist breadth of men is equal to 9 cm, we will use a one-sample z-test.

The null hypothesis is: H0: µ = 9 cm

The alternative hypothesis is: Ha: µ ≠ 9 cm

We are given a sample of n = 72 men with a sample mean of x = 8.91 cm and a population standard deviation = 0.36 cm.

The test statistic for a one-sample z-test is given by: z = (x - µ) / (o / sqrt(n))

Substituting the given values, we get:

= (8.91 - 9) / (0.36 / sqrt(72))

z = -2.119

At a significance level of a = 0.01, the critical values for a two-tailed test are ±2.576.

Since our test statistic (-2.119) falls outside of this range, we reject the null hypothesis and conclude that there is sufficient evidence to suggest that the mean wrist breadth of men is not equal to 9 cm.

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The height of the cylinder is 3.6 inches.

We know that the volume of a cylinder of radius R and height H is:

V = pi*R²*H

where pi = 3.14

We know that the radius is R = 5in and the volume is 284.7 inches cubed, replacing that in the formula above we will get:

284.7 in³= 3.14*(5 in)²*H

Solving that for H we will get:

H= (284.7 in³)/ 3.14*(5 in)²

H = 3.6 inches.

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The function represented by the data is f(1/4)x³ + 2x² - 24. The correct option is 2.

In the given table, we have the values of x and f(x) for x=0,1,2,3, and 4. We need to find a polynomial function that satisfies these data points.

Looking at the table, we can see that f(x) is negative for x=0,1,2 and positive for x=3,4. This suggests that the polynomial has a root or a zero between x=2 and x=3.

To find the degree of the polynomial, we count the number of data points given. Since we have 5 data points, we need a polynomial of degree 4.

We can use interpolation to find the coefficients of the polynomial. One way to do this is to set up a system of equations using the data points:

f(0) = -24 = a(0)⁴ + b(0)³ + c(0)² + d(0) + e

f(1) = -21.75 = a(1)⁴ + b(1)³ + c(1)² + d(1) + e

f(2) = -14 = a(2)⁴ + b(2)³ + c(2)² + d(2) + e

f(3) = 0.75 = a(3)⁴ + b(3)³ + c(3)² + d(3) + e

f(4) = 24 = a(4)⁴ + b(4)³ + c(4)² + d(4) + e

Solving this system of equations gives us the polynomial function:

f(x) = -0.25x⁴ + 2x³ - 2.75x² - 0.5x + 24

Therefore, the correct option is 2. f(x) = (1/4)x³ + 2x² - 24.

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