Create trig ratios for sin, cos, and tan:

ans.(a) is correct

only in option (a) numbers are arranged from least to greatest.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

A probability can be classified as experimental or theoretical, as follows:

Over a small number of trials, these two probabilities can be different, but over a large number of trials, their values get closer.

The dice has eight sides, hence the theoretical probability of rolling a six is given as follows:

1/8 = 0.125 = 12.5%.

(each of the eight sides is equally as likely, and a six is one of these sides).

The experimental probabilities are obtained considering the trials, hence:

(given in the problem).

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Answer:

x = -9

Step-by-step explanation:

As this is an absolute value function, the line of symmetry is the x-value of the maximum/minimum point. An absolute value function can be denoted as y = |x - h| + k, where (h, k) is the maximum/minimum point. We only need the x-value of the maximum/minimum point, so we only have to look at "h". Now, we can use y = |x - h| + k and turn it into g(x):

y = |x - h| + k

g(x) = |x - -9| + -11   --> this means h = -9, and the line of symmetry is at x = -9

g(x) = |x + 9| - 11

Answer:

I think x equals --9

Answer:

5x=380

x = 76

Carla made 76 cookies each hours

Step-by-step explanation:

Just make an equation, so the total number of cookies is 380 and she works for 5 hours, so it is just 380/5.

Answer:

1/221.

Step-by-step explanation:

Probability(first card is a King) = 4/52 = 1/13 (as there are 4 kings in the pack).

Now there are 51 cards left in the pack, 3 of which are Kings, so:

Probability(second card is a King) = 3/51 = 1/17.

These 2 events are independent so we multiply the probabilities:

Required probability =

1/13 * 1/17

= 1/221.

dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

To find dx/dy, we first need to find dx/dt and dy/dt, and then we can use the chain rule.

Given x = sin^3(t) and y = cos^3(t),

dx/dt = d(sin^3(t))/dt = 3sin^2(t) * cos(t) (using the chain rule)
dy/dt = d(cos^3(t))/dt = -3cos^2(t) * sin(t) (using the chain rule)

Now, we can find dx/dy by dividing dx/dt by dy/dt:

dx/dy = (dx/dt) / (dy/dt) = (3sin^2(t) * cos(t)) / (-3cos^2(t) * sin(t))

Simplify the expression:

dx/dy = -sin(t)/cos(t)

So, dx/dy = -sin(t)/cos(t) when x = sin^3(t) and y = cos^3(t).

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To determine the measure of the unknown segment, it's essential to first gather information about the given problem, such as the context, any provided measurements, and any relationships between the segments or angles involved. Once you have this information, you can utilize relevant geometric principles and theorems to establish connections and solve for the unknown value.

For example, if the unknown segment is a side in a triangle, you may apply the Pythagorean theorem, triangle inequality theorem, or trigonometric functions such as sine, cosine, or tangent to calculate its length. If the unknown segment is part of a circle, you might use the properties of arcs, chords, or the circumference to determine its measure. In cases where the unknown segment is part of a polygon, you can consider properties like diagonals, perimeter, or area to derive its length.

After identifying the appropriate method and relationships, you can set up equations and solve for the unknown variable. To verify the solution, you can plug it back into the original problem to ensure it satisfies all given conditions. In conclusion, finding the measure of an unknown segment involves understanding the problem's context, applying relevant geometric concepts, and using mathematical techniques to solve for the desired value.

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Answer:

  C.  Box B

Step-by-step explanation:

You want to know which of these two boxes has a volume greater than 9.5 ft³:

The volume of each box is found by multiplying its dimensions:

  Box A: (3 ft)(2 ft)(1.5 ft) = 9 ft³

  Box B: (2.5 ft)(2 ft)(2 ft) = 10 ft³

Only box B is large enough, choice C.

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To determine whether a critical point is a relative minimum or maximum using concavity, we need to examine the second derivative of the function at the critical point.

If the second derivative is positive, then the function is concave up, meaning it is shaped like a bowl opening upwards. At a critical point where the first derivative is zero, this indicates a relative minimum, as the function is increasing on either side of the critical point.

On the other hand, if the second derivative is negative, then the function is concave down, meaning it is shaped like a bowl opening downwards. At a critical point where the first derivative is zero, this indicates a relative maximum, as the function is decreasing on either side of the critical point.

If the second derivative is zero, then the test is inconclusive and further analysis is needed, such as examining higher order derivatives or using other methods such as the first derivative test.

Therefore, the concavity test is a useful method for determining the nature of critical points and whether they represent a relative minimum or maximum.

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Complete question is:

What we need to examine for a method for determining whether a critical point is a relative minimum or maximum using concavity.

According to given question Mike has 25 candies in the beginning.

Let's assume that Mike had "x" candies in the beginning.

After giving 15 candies to his friend, he would have had (x - 15) candies left.

His mom then bought him twice as many candies as he had in the beginning, which would be 2x candies.

So, the total number of candies Mike has now is (x - 15) + 2x = 60.

Combining like terms, we get 3x - 15 = 60.

Adding 15 to both sides, we have 3x = 75.

Finally, dividing both sides by 3, we find that x = 25.

Therefore, Mike had 25 candies in the beginning.

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The complete question is  Mike had some candies. He gave 15 of them to his friend. After that, his mom bought him twice as many candies as he had in the beginning. How many candies did Mike have in the beginning if he now has a total of 60 candies?

Prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

Let's first understand the Mean Value Theorem (MVT). The MVT states that if a function is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that:

f'(c) = (f(b) - f(a)) / (b - a)

Now, consider the function f(x) = 1/x^2. This function is continuous and differentiable for all x ≠ 0. However, in the interval [-1, 1], the function is not continuous nor differentiable at x = 0. Therefore, the Mean Value Theorem does not apply to this interval.

Since the MVT does not apply, we cannot say there exists a c in the interval (-1, 1) such that f'(c) = (f(1) - f(-1)) / (1 - (-1)). This is because the prerequisites for the MVT are not met due to the discontinuity and non-differentiability of f(x) at x = 0 within the interval [-1, 1].

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Dina can reach a maximum kinetic energy of 2450 Joules when she skis to the bottom of the slope.

Potential energy (PE) = m x g x h

where m = mass, g = acceleration due to gravity, and h = height

Here, Dina's mass (m) = 50 kg, height (h) = 5 m, and acceleration due to gravity (g) = 9.8 m/s².

So, PE = 50 x 9.8 x 5

PE = 2450 Joules

When Dina skis down the slope, all of her potential energy will be converted into kinetic energy (KE) at the bottom of the slope, neglecting any losses due to friction and air resistance.

Thus, the maximum kinetic energy (KE) Dina can reach at the bottom of the slope is also 2450 Joules.

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The maximum kinetic energy she can reach when she skis to the bottom of the slope is 2452 joules

To find the maximum kinetic energy that Dina can reach when she skis to the bottom of the slope, we need to use the principle of conservation of energy, which states that the total energy of a system remains constant.

At the top of the slope, Dina has potential energy due to her position relative to the ground. This potential energy is given by:

PE = m × g × h

where m is Dina's mass (50 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slope (5 m).

PE = 50 kg × 9.8 m/s^2 × 5 m = 2450 J

When Dina skis to the bottom of the slope, all of her potential energy is converted into kinetic energy, which is given by:

KE = 1/2 × m × v^2

where v is her velocity at the bottom of the slope.

To find the maximum velocity, we can use the fact that the total energy of the system remains constant:

PE = KE

2450 J = 1/2 × 50 kg × v^2

v^2 = 98 m^2/s^2

v = sqrt(98) = 9.90 m/s

Finally, we can substitute this velocity into the kinetic energy equation to find the maximum kinetic energy that Dina can reach:

KE = 1/2 × 50 kg × (9.90 m/s)^2 = 2452 J

Therefore, Dina can reach a maximum kinetic energy of 2452 Joules when she skis to the bottom of the slope.

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The median means that as many as friends have less than A. 1. 5 pets as those that have more than A. 1. 5 pets.

The median is a measure of central tendency in statistics that represents the middle value of a dataset when it is ordered from smallest to largest. The median is often used as a more robust measure of central tendency than the mean, because it is less affected by extreme values in the dataset.

From the box plot, the data set of friends with pets would be:

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, ,2 ,2, 2, 2, 2, 3, 4, 4, 7.

The median here is:

= ( 10 th position + 11 th position ) / 2

= ( 1 + 2 ) / 3

= 1. 5

This therefore means that as many friends have more than 1. 5 pets as those with less than 1. 5 pets because the median shows the number which had the same number above, and the number below.

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The surface area of the composite figure is given as follows:

446 cm².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The figure is this problem is given by the composition of two rectangular prisms, with dimensions given as follows:

Hence the surface area is given as follows:

S = 2 x (5 x 10 + 5 x 7 + 10 x 7) + 2 x (8 x 4 + 8 x 3 + 4 x 3)

S = 446 cm².

The prism is given by the image presented at the end of the answer.

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The missing side length is 5√3 centimeters.

We can use the Pythagorean theorem to find the missing side length. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. In equation form, this looks like:

a² + b² = c²

where a and b are the lengths of the legs, and c is the length of the hypotenuse.

To use this formula to solve for the missing side length, we can plug in the values we know:

5² + b² = 10²

We can simplify this equation by squaring 5 and 10:

25 + b² = 100

Next, we can isolate the variable (b) on one side of the equation by subtracting 25 from both sides:

b² = 75

Finally, we can solve for b by taking the square root of both sides:

b = √(75)

This simplifies to:

b = 5*√(3)

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Complete Question:

By using the Pythagoras theorem, Find the value of the Other side when the value of hypotenuse is 10 cm and the value of the side is 5 cm.

m∠f to the nearest degree is 83°.

The Law of Cosines is used to find an angle when all triangle sides are known.

f² = d² +e² -2de cos(F)

To find angle ∠F, we need to find the value of cos(∠F), which we can do by rearranging the Law of Cosines as follows:

cos(F) = (d² +e² -f²) / (2de)

cos(F) = (20² + 25² - 30²) / (2 × 20 × 25)

cos(F) = (400 + 625 - 900) / (1000)

cos(F) = 125/1000

∠F = arccos(1/8)

∠F = 82.8°

Rounding to the nearest degree

∠F = 83°

Hence, m∠f to the nearest degree is 83°.

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a. The smallest value in the data set is 4 miles per hour. b. The largest value in the data set is 48 miles per hour. c. The median is 26 miles per hour.

a. The smallest value in the data set is 4 miles per hour.

b. The largest value in the data set is 48 miles per hour.

c. To find the median, we need to arrange the values in order from smallest to largest:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

The median is the middle value in this list. Since there are an even number of values, we take the average of the two middle values:

Median = (24 + 28) / 2 = 26

Therefore, the median speed of the 100 cars as they passed through the busy intersection is 26 miles per hour.

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To evaluate the indefinite integral of 3x^2 – 3x +1/ x^3 + 2, we can use partial fraction decomposition.

First, we factor the denominator: x^3 + 2 = (x + ∛2)(x^2 – ∛2x + 2).

Next, we can write the fraction as:

3x^2 – 3x +1/ x^3 + 2 = A/x + B(x^2 – ∛2x + 2) + C(x + ∛2)

Multiplying both sides by the denominator, we get:

3x^2 – 3x + 1 = A(x^2 – ∛2x + 2)(x + ∛2) + Bx(x + ∛2) + C(x^2 – ∛2x + 2)

To solve for A, B, and C, we can plug in specific values of x. For example, if we plug in x = -∛2, we get:

-2√2 + 1 = A(4√2) + C(0)

Therefore, A = (2 – √2)/8.

If we plug in x = 0, we get:

1 = A(2√2) + B(0) + C(√2)

Therefore, C = 1/√2.

Finally, if we plug in x = 1, we get:

1 = A(3√2) + B(1 – √2) + C(1 + √2)

Therefore, B = (-1 + √2)/4.

Now that we have A, B, and C, we can write the original fraction as:

3x^2 – 3x +1/ x^3 + 2 = (2 – √2)/8 * 1/x + (-1 + √2)/4 * (x^2 – ∛2x + 2) + 1/√2 * (x + ∛2)

Using this partial fraction decomposition, we can now integrate each term separately.

Integrating the first term, we get:

∫(2 – √2)/8 * 1/x dx = (1/8)(2ln|x| – √2 ln|x^2 + 2|) + C

Integrating the second term, we can complete the square to get:

∫(-1 + √2)/4 * (x^2 – ∛2x + 2) dx = (-1 + √2)/4 * ∫(x – ∛1/2)^2 + 3/2 dx = (-1 + √2)/4 * ((x – ∛1/2)^2 + 3/2) + C

Integrating the third term, we get:

∫1/√2 * (x + ∛2) dx = (1/2√2) * (x^2/2 + ∛2x) + C

Putting it all together, we have:

∫(3x^2 – 3x +1)/ (x^3 + 2) dx = (1/8)(2ln|x| – √2 ln|x^2 + 2|) + (-1 + √2)/4 * ((x – ∛1/2)^2 + 3/2) + (1/2√2) * (x^2/2 + ∛2x) + C

where C is the constant of integration.

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The basal area of the tree after 10 years of growth would be approximately 279.7 square inches.

Given the initial basal area of a tree, which is 154 square inches, and the annual growth rate, which is 6%. To find out what the basal area of the tree will be after 10 years of growth.

By using the formula for compound interest, which can be applied to the growth of the basal area over time. The formula is:

A = P(1 + r)ⁿ

where:

A is the final amount

P is the initial amount

r is the annual growth rate

n is the number of years

To find A, the final basal area of the tree after 10 years of growth. We know that P is 154 square inches, r is 6% or 0.06 and n is 10.

By applying these values in the formula, we get:

A = 154(1 + 0.06)¹⁰

A = 154(1.06)¹⁰

A = ≈ 279.7

Therefore, the basal area of the tree after 10 years of growth would be approximately 279.7 square inches.

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The price of one apple is $0.70, obtained by solving the system of equations 4x + 9y = 12.70 and 8x + 11y = 17.70 using elimination.

Let's use a system of equations caculation the problem.

Let x be the price of one apple and y be the price of one banana.

From the first sentence, we know that:

4x + 9y = 12.70

From the second sentence, we know that:

8x + 11y = 17.70

Now we can solve for x by using either substitution or elimination.

Let's use elimination.

We can multiply the first equation by 11 and the second equation by -9, then add them together:

44x + 99y = 139.70

-72x - 99y = -159.30

-28x = -19.60

Dividing both sides by -28, we get:

x = 0.70

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The measure of angle OAC if major arc AB measures 220 degrees is 110 degrees. Therefore, the correct option is C.

To find the measure of angle OAC, we need to use the central angle theorem which states that the measure of an inscribed angle is equal to half the measure of the intercepted arc.

Here, we are given that the major arc AB measures 220 degrees. So, the measure of angle AOB (the central angle) is 220 degrees.

Since angle OAC is an inscribed angle that intercepts arc AB, its measure is half the measure of arc AB.

Therefore, measure of angle OAC = (1/2) * 220 = 110 degrees.

So, the correct answer is option (c) 110.

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the evaluated double integral is approximately 14.25.

To evaluate the given double integral, we need to first understand the problem properly. We have the function f(x, y) = y + xy, and the region R is described by the inequalities: 0 < x < y^2, and 1 < y < 2.

Now we can set up the double integral:

∬(y + xy) dA over the region R.

Since we are given that 0 < x < y^2 and 1 < y < 2, we can set up the integral using the given limits of integration:

∫(from y = 1 to 2) ∫(from x = 0 to y^2) (y + xy) dx dy.

Now, we can start by integrating the inner integral with respect to x:

∫(from y = 1 to 2) [(yx + (1/2)x^2*y) evaluated from x = 0 to x = y^2] dy.

After evaluating the inner integral, we have:

∫(from y = 1 to 2) (y^3 + (1/2)(y^2)^2*y) dy.

Now, we can integrate the outer integral with respect to y:

[((1/4)y^4 + (1/6)y^6) evaluated from y = 1 to y = 2].

After evaluating the outer integral, we get:

[(1/4)(2^4) + (1/6)(2^6)] - [(1/4)(1^4) + (1/6)(1^6)].

Calculating the final result:

(4 + 10.6667) - (0.25 + 0.1667) = 14.6667 - 0.4167 ≈ 14.25.

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The number line that represents the solution to this inequality is 6.10, with an open circle at 25 and shading to the right.

To solve the inequality 12t > 300, we need to isolate t on one side of the inequality. We can do this by dividing both sides by 12:

12t/12 > 300/12

t > 25

This means that Tina must sell more than 25 tickets in order to meet her goal of selling more than $300 worth of tickets.

To represent this solution on a number line, we can start by plotting a point at 25. Since the inequality is greater than (>) and not greater than or equal to (≥), we use an open circle at 25.

Then, we need to shade the area to the right of 25 to represent all the possible values of t that satisfy the inequality. This is because any value of t greater than 25 will make 12t greater than 300.

Out of the answer choices given, the number line that represents the solution to this inequality is 6.10, with an open circle at 25 and shading to the right.

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To find the temperature at 6pm, we need to start with the temperature at 9am and then add or subtract the changes in temperature that occurred during the day.

We know that the temperature at 9am was -1.9°F. Between 9am and noon, the temperature rose 11.3°F, so at noon the temperature was:

-1.9 + 11.3 = 9.4°F

Between noon and 3pm, the temperature dropped 7.9°F, so at 3pm the temperature was:

9.4 - 7.9 = 1.5°F

Between 3pm and 6pm, the temperature dropped 12.7°F, so at 6pm the temperature was:

1.5 - 12.7 = -11.2°F

Therefore, the temperature at 6pm was -11.2°F.

The expression that is equivalent to 3v+2v is:

D. v+v+v+v+v

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we substitute the same value(s) for the variable(s).

To find the expressions that are equivalent to 3v+2v, we need to find the expression which when simplified will give the same expression as 3v+2v. That is: 3v + 2v = 5v

v*5 = 5v

v+5 = v + 5

v+5v = 6v

v+v+v+v+v = 5v

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The point (2, 8) is the point (x1, y1) identified from the equation y - 8 = 3(x - 2

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Answer:

------------------------------

The two smaller right triangles are similar by AA property.

Use ratios of corresponding sides to get:

Simplify and solve for y:

Answer:

x/4 + 3/28 = y

Step-by-step explanation:

To find the inverse, switch the x's and y's (note that f(x) is y) and solve for y:

x = 4y - 3/7

x + 3/7 = 4y

x/4 + 3/28 = y

Answer:

−1(x) = 3√2(x+7) 2 f - 1 (x) = 2 (x + 7) 3 2 is the inverse of f (x) = 4x3 − 7 f (x) = 4 x 3 - 7.

The distance from Xavier and David to the ranch can be found using the distance formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

In this case, we are given the distances DR and XR, which represent the change in x, y, and z coordinates between the tents and the ranch. We know that the tents are located at equal distances from the ranch, so the change in x, y, and z coordinates for both David and Xavier will be the same.

Let's call the distance from each tent to the ranch "d", then we have:

DR = XR = d

Substituting the given values, we get:

12.3z + 12.4 = 10.5z + 34

Solving for z, we get:

z = 6.8

Now we can find the distance from each tent to the ranch using the formula:

distance = sqrt((change in x)^2 + (change in y)^2 + (change in z)^2)

For David's tent:

distance = sqrt((12.3z)^2 + 0^2 + (12.4)^2) = sqrt((12.3*6.8)^2 + (12.4)^2) = 87.9 meters (rounded to one decimal place)

For Xavier's tent:

distance = sqrt((10.5z)^2 + 0^2 + (34)^2) = sqrt((10.5*6.8)^2 + (34)^2) = 95.6 meters (rounded to one decimal place)

Therefore, David and Xavier are 87.9 meters and 95.6 meters away from the ranch, respectively.