Unit 7: Right Triangles & Trigonometry Homework 4: Trigonometry Ratios & Finding Missing Sides #’s 10&11

Answer:

when the diameter of a spherical balloon shrinks to one-half of its original size, the surface area of the balloon will decrease to one-fourth of its original size, while the volume will decrease to one-eighth of its original size.

Step-by-step explanation:

When the diameter of a spherical balloon shrinks to one-half of its original size, it means the new diameter is half of the original diameter.

Let's assume the original diameter of the balloon was "d," then the new diameter will be "d/2."

Surface Area:

The surface area of a sphere is given by the formula 4πr², where "r" is the radius of the sphere. Since the diameter of the sphere is halved, the radius will also be halved. Therefore, the new surface area (SA') of the balloon will be:

SA' = 4π (d/4)² = πd²/4

Thus, the new surface area of the balloon will be one-fourth (1/4) of the original surface area.

Volume:

The volume of a sphere is given by the formula (4/3)πr³. Again, since the diameter of the sphere is halved, the radius will also be halved. Therefore, the new volume (V') of the balloon will be:

V' = (4/3)π (d/4)³ = πd³/24

Thus, the new volume of the balloon will be one-eighth (1/8) of the original volume.

The critical values are x = -4 and x = 2.

Given the function f(x) = (2-x)(x+4)^2, we need to find the critical values.

Critical values are the points where the derivative of the function is either zero or undefined.

Step 1: Find the derivative of f(x). f'(x) = d/dx((2-x)(x+4)^2)

Step 2: Apply the product rule, which states d(uv) = u*dv + v*du,

where u = (2-x) and v = (x+4)^2. f'(x) = (2-x)*d/dx((x+4)^2) + (x+4)^2*d/dx(2-x)

Step 3: Compute the individual derivatives. f'(x) = (2-x)*(2(x+4)) + (x+4)^2*(-1)

Step 4: Simplify the expression. f'(x) = -2(x+4)^2 + 4(x+4)(2-x)

Step 5: Set f'(x) equal to 0 and solve for x. 0 = -2(x+4)^2 + 4(x+4)(2-x)

Step 6: Factor out a common term. 0 = 2(x+4)[-1(x+4) + 2(2-x)]

Step 7: Solve for x. 0 = 2(x+4)(-x+2) x = -4, 2 The critical values are x = -4 and x = 2.

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To find f'(4) for the function f(x) = ln(2x^3), we first need to find the derivative f'(x) using the chain rule.

The chain rule states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function.

For f(x) = ln(2x^3), the outer function is ln(u) and the inner function is u = 2x^3.

The derivative of the outer function, ln(u), is 1/u.
The derivative of the inner function, 2x^3, is 6x^2 (using the power rule).

Now, apply the chain rule: f'(x) = (1/u) * 6x^2 = (1/(2x^3)) * 6x^2.

Simplify f'(x): f'(x) = 6x^2 / (2x^3) = 3/x.

Now, find f'(4): f'(4) = 3/4.

So, f'(4) for f(x) = ln(2x^3) is 3/4 or 0.75 when rounded to 2 decimal places.

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The number sequence that follows the rule of subtracting 15 starting from 105 is 105, 90, 75, 60, 45.

To obtain this sequence, we start with 105 and subtract 15 from it to get 90. Then we subtract 15 from 90 to get 75, and so on until we reach 45. Each term in the sequence is obtained by subtracting 15 from the previous term.

It is important to note that this is an arithmetic sequence with a common difference of -15. The formula for finding the nth term of an arithmetic sequence is: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, and d is the common difference. Using this formula, we can find any term in the sequence by plugging in the appropriate values.

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There are 120 possible orders in which the five Smith children can be served one at a time.

If we assume that each child is served one at a time, then the first child can be served in 5 ways, the second child can be served in 4 ways (since one child has already been served), the third child can be served in 3 ways, the fourth child can be served in 2 ways, and the fifth child can be served in 1 way.

Therefore, the total number of orders in which the children can be served one at a time is:

5 x 4 x 3 x 2 x 1 = 120

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Answer: b = 30°

Step-by-step explanation:

       The little square represents a 90-degree angle.

90° - 60° = b

30° =  b

b = 30°

la probabilidad de que Antawn Jamison anote sus siguientes 9 tiros libres es del 7.33%.

What is probability?

By simply dividing the favorable number of possibilities by the entire number of possible outcomes, the probability of an occurrence can be determined using the probability formula. Because the favorable number of outcomes can never exceed the entire number of outcomes, the chance of an event occurring might range from 0 to 1.

La probabilidad de que anote un tiro libre es del 73%, lo que significa que la probabilidad de que falle es del 27%.

La probabilidad de que anote sus próximos 9 tiros libres es:

0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 x 0.73 = 0.0733

Por lo tanto, la probabilidad de que Antawn Jamison anote sus siguientes 9 tiros libres es del 7.33%.

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For March 2020, the equivalent units of production for conversion costs are 30,750, rounded to the nearest whole number.

To determine the equivalent units of production for conversion costs in March 2020 for Dorchester Company, we need to follow these steps:
Calculate the equivalent units for the beginning work in process inventory:
Beginning WIP units = 6,000
Completion percentage for conversion costs = 50%
Equivalent units for beginning WIP = 6,000 * 50% = 3,000
Calculate the equivalent units for the units started and completed during March:
Units started and completed = 24,500
Completion percentage for conversion costs = 100%
Equivalent units for started and completed units = 24,500 * 100% = 24,500
Calculate the equivalent units for the ending work in process inventory:                                                                         Ending WIP units = 6,500
Completion percentage for conversion costs = 50%
Equivalent units for ending WIP = 6,500 * 50% = 3,250
Sum up the equivalent units from steps 1, 2, and 3:
Total equivalent units for conversion costs = 3,000 (beginning WIP) + 24,500 (started and completed units) + 3,250 (ending WIP) = 30,750
So, for March 2020, the equivalent units of production for conversion costs are 30,750, rounded to the nearest whole number.

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It appears that the relationship between the number of registered pleasure boats and manatee deaths fluctuates over the years. While there is no clear trend, it is important to consider the possible effects of increased boat registration on manatee populations.

The table you provided shows the number of registered pleasure boats (in tens of thousands) and the number of manatee deaths caused by boats in Florida from 1991 to 2014.

When the number of registered pleasure boats increases, there could be a higher likelihood of boat-related manatee deaths. As more boats are present in the water, manatees may face increased risks from boat strikes, which can lead to injuries or fatalities. Additionally, more boats may lead to habitat destruction, indirectly affecting manatee populations.

It is crucial for boat owners to follow safe boating practices to protect manatees and their habitats. Some measures to reduce manatee deaths include obeying speed limits in designated manatee zones, wearing polarized sunglasses to increase visibility, and being cautious in shallow areas where manatees might be feeding or resting.

In conclusion, the relationship between the number of registered pleasure boats and manatee deaths in Florida is complex and fluctuating. However, it is essential for boat owners to be aware of the potential risks and take necessary precautions to protect these gentle marine mammals.

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The type of triangle formed from the side lengths of a triangle 2, 10, 11 is a scalene triangle.

Based on the given side lengths of a triangle (2, 10, 11), we can determine the type of triangle formed by examining their relationships. First, let's check if these side lengths can form a valid triangle using the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the remaining side. In this case, 2 + 10 > 11, 2 + 11 > 10, and 10 + 11 > 2, so a triangle can be formed.

Now let's identify the type of triangle. There are three main categories to consider: equilateral, isosceles, and scalene. An equilateral triangle has all three sides equal in length, which doesn't apply here. An isosceles triangle has two sides with equal lengths, but in this case, all three sides have distinct lengths. Therefore, the triangle is a scalene triangle, meaning it has no sides of equal length.

In summary, the triangle formed by side lengths 2, 10, and 11 is a scalene triangle because all three sides have different lengths and satisfy the Triangle Inequality Theorem.

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The quadratic formula for solving quadratic equations of the form ax^2 + bx + c = 0 is:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

To put the quadratic equation y = 2x^2 - 4x + 2 into this formula, we need to identify the values of a, b, and c.

In this case, we have a = 2, b = -4, and c = 2. Substituting these values into the quadratic formula, we get:

x = (-(-4) ± sqrt((-4)^2 - 4(2)(2))) / (2 * 2)

x = (4 ± sqrt(16 - 16)) / 4

x = (4 ± 0) / 4

Simplifying this expression, we get:

x = 1 or x = 1/2

Therefore, the solutions to the quadratic equation y = 2x^2 - 4x + 2 are x = 1 and x = 1/2.

To explain this solution in more detail, we first need to understand the quadratic formula and how it can be used to solve quadratic equations. The quadratic formula is a formula that provides the solutions to any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are constants.

In this case, we were given a specific quadratic equation, y = 2x^2 - 4x + 2, and we needed to find its solutions. To do this, we identified the values of a, b, and c and substituted them into the quadratic formula. We then simplified the expression to obtain the solutions, which were x = 1 and x = 1/2.

It is important to be able to use the quadratic formula to solve quadratic equations because many real-world problems can be modeled using quadratic equations. By being able to solve these equations, we can find important information such as the roots, or solutions, of the equation, which can help us make predictions and solve problems.

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Answer:

2

Step-by-step explanation:

The median of a data set is the middle value when the data is arranged in order of size.

The given table of data is:

[tex]\begin{array}{|c|c|c|c|}\cline{1-4}\vphantom{\dfrac12}\frac{3}{4}&3&1&8\\\cline{1-4}\vphantom{\dfrac12}5&\frac{1}{4}&\frac{3}{5}&\frac{7}{2}\\\cline{1-4}\end{array}[/tex]

Arrange the data in order of size:

[tex]\dfrac{1}{4},\;\dfrac{3}{5},\;\dfrac{3}{4},\;1,\;3,\;\dfrac{7}{2},\;5,\;8[/tex]

As there are 8 data values, the median is the average of the two middle values.

The two middle values are 1 and 3.

The average of 1 and 3 is 2:  

[tex]\dfrac{1+3}{2}=\dfrac{4}{2}=2[/tex]

Therefore, the median of the given data is 2.

The correct equation is,

⇒ (5 - 6i) (2 + 4i)

Let us assume that;

⇒ (a + bi) (c + di)

⇒ ac + adi + bci + bdi²

⇒ (ac − bd) + (ad + bc)i

Matching coefficients:

30 < ac − bd < 80

ad + bc = 0

Hence, We need to pick four integers between -9 and 9 such that these two equations are satisfied.  One possible combination is:

a = 8, b = -4, c = 6, d = 3

The number would be:

ac − bd = (8)(6) − (-4)(3) = 60

Puzzle 2

Using the result from Puzzle 1:

ac − bd = 34

ad + bc = 8

Like before, it makes sense to assume b is negative.  With some trial and error, one possible answer is:

a = 5, b = -6, c = 2, d = 4

Thus, The correct equation is,

⇒ (5 - 6i) (2 + 4i)

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For the function f(x, y) = 1 - xy + y²= -y = -1, fy(-2,1) = 2y - x = 3.

When y is stationary, the rate of change of f with respect to x is determined by the partial derivative fx. Meanwhile, fy measures the rate of change of f in relation to y while holding x constant.

At the point (-2,1), the fx partial derivative symbolizes the incline of the level curve tangent line that follows the x-direction in relation to f. Its counterpart fy denotes the incline of the level curve tangent line that follows the y-direction regarding f at (-2,1).

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Complete answer - Let f(x, y) = 1 - xy + y² Compute fx(-2, 1) and fy(-2,1), and interpret these partial derivatives geometrically."

Answer:

measure of angle GHJ = 1/2 the measure of arc GJ = (1/2)(86°) = 43°

measure of angle JIG = 1/2 the measure of arc GJ = (1/2)(86°) = 43°

The length of UX (the length of a tangent segment to a circle) is approximately 4.9 inches.

To find the length of UX, we can use the formula for the length of a tangent segment to a circle:

Length of tangent segment = √(radius² - distance from center²)

In this case, we don't know the radius or the distance from the center, but we can use the fact that RU is perpendicular to UT to find them:

RU = RS + ST = 8 + 4 = 12 in.
UT = radius = RU/2 = 12/2 = 6 in.

Now we can plug these values into the formula:

Length of tangent segment = √(6² - 4²) ≈ 4.9 in.

Therefore, the length of UX is approximately 4.9 inches.

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Answer: Regarding "Car Mr V Model 2017," I'm not sure what you're asking. Can you please provide more context or a clear question?

Step-by-step explanation:

To determine if Eugene's thinking is correct, we need to calculate his new monthly salary with the 3.5% raise.

3.5% of $2,700 is (3.5/100) x $2,700 = $94.50

Eugene's new monthly salary is $2,700 + $94.50 = $2,794.50

So, Eugene's thinking is not correct. His new monthly salary with the 3.5% raise is $2,794.50, which is still less than $2,800.

Today's lesson was not provided in your question. Please provide a topic or question for me to provide a summary of today's lesson.

Regarding "Car Mr V Model 2017," I'm not sure what you're asking. Can you please provide more context or a clear question?

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The two supplied points (1, 4) and (1, −7), have identical x-values. These would create a vertical line if graphed, then joined.

The points where the lines representing the intersections where the two linear equations intersect are referred to as the solution of a linear equation.

Given that:

Line has points:  (1, 4) and (1, −7).

The two supplied points have identical x-values. These would create a vertical line if graphed, then joined.

The diagram is attached:

Hence, a vertical line would be formed by the points.

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Complete question:

Which of the following is true of (1, 4) and (1, −7).?

options:

The polynomial expression becomes -20vz⁶ + 36v⁴z⁵ + 24v⁶ z⁶

In order to divide, first collect the like terms and then perform the operation. So, we have:

(-20vz⁶ + 32v⁴z⁵ +24v⁶ z⁶) + (4v⁴ z⁵) = -20vz⁶ + (32v⁴z⁵ +4v⁴ z⁵) + 24v⁶ z⁶

Simplifying the expression in parentheses, we get:

32v⁴z⁵ +4v⁴ z⁵ = 36v⁴z⁵

So, the expression becomes:

-20vz⁶ + 36v⁴z⁵ + 24v⁶ z⁶

This expression cannot be simplified any further.

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Answer:

Step-by-step explanation:

The number of favorable outcomes for spinning the same color twice will depend on the number of colors on the spinner.

If there are only two colors on the spinner, such as red and blue, then there will be only one favorable outcome, which is spinning either red or blue twice.

If there are more than two colors on the spinner, the number of favorable outcomes will depend on the number of times each color appears on the spinner.

For example, if there are four colors on the spinner, and each color appears equally, then there will be four favorable outcomes: spinning red twice, spinning blue twice, spinning green twice, or spinning yellow twice.

In general, if there are n colors on the spinner and each color appears with equal probability, then the number of favorable outcomes for spinning the same color twice will be n.

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The value of x using the assumptions of angle are 17 and 22.5

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

Angles: 75° and (4x + 7)°

Angles: (2x+10)° and (2x - 10)°

Assume 75° and (4x + 7)° are congruent angles, we have

4x + 7 = 75

So, we have

4x = 68

Divide by 4

x = 17

Assume (2x+10)° and (2x - 10)° are complementary angles, we have

2x + 10 + 2x - 10 = 90

So, we have

4x = 90

Divide by 4

x = 22.5

Hence, the values of x are 17 and 22.5

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The final answer to this limit question is 1/4.

a function from a set X to a set Y assigns to each element of X exactly one element of Y.[1] The set X is called the domain of the function[2] and the set Y is called the codomain of the function.

Given that lim f(x) = 3 and lim g(x) = 12, we want to find the limit:
lim (f(x) / g(x)) as x approaches -5.
Using the limit rules, specifically the quotient rule, we have:
lim (f(x) / g(x)) = lim f(x) / lim g(x)

Now, substituting the given limits:
lim (f(x) / g(x)) = 3 / 12

Simplifying the fraction:

lim (f(x) / g(x)) = 1/4

So, the answer is 1/4.

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Answer:

Recall 5%=.05 as a decimal. Then multiply 140(.05)=7 to get how many centimeters he grew. Add 140+7=147 to get his current height.

The slope of the line represent Parking cost $30 per day. The correct answer is C.

The slope of a line represents the rate of change between two variables. In this case, the line represents the relationship between Hector's parking fees and the number of days he parked his SUV.

The unit of the slope of the ine represents

Cost/ number of days = 30 $/day

The fact that the slope is negative (-$30) means that for each additional day Hector parked his SUV, his parking fees decreased by $30. This indicates that the parking fee is a function of the number of days parked, and it costs $30 per day. The correct option is C.

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--The given question is incomplete, the complete question is given

" Hector parked his SUV in long-term parking while he traveled to China. What does the slope of the line represent?

A Hector's parking fees decreased by $30 each week.

B Parking cost a flat fee of $30.

C Parking cost $30 per day.

D Hector got a $30 discount to park his SUV. ​ "--

f(0.5) represents the volume of the smaller box when its' side length is 0.5 cm less than the larger cube

The formula to find the Volume of a cube is:

V = x³

where:

x is the side length of the cube

The cube has a volume of 27 cm³. Thus:

Side length of cube = ∛27 = 3 cm

Since the side length of the smaller cube is x cm less than side length of the larger cube, then we can say that the function to find the volume of the smaller cube is:

V_small: f(x) = (3 - x)³

Thus, f(0.5) is the volume of the smaller box when its' side length is 0.5 cm less than the larger cube

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Answer: x(squared) + (y + 2)(squared) = 36

Step-by-step explanation:

The center of the circle is at (0 , -2), so we will eliminate the x value by just writing x(squared), and then we will write the y value as (y + 2)(squared) because the standard form for a circle is

(x - h)(squared) + (y + 2)(squared) = r(squared)

And if we count from the center to the top point of the circle, the number would be 6, so we would square 6 which would be 36.

The pollster should survey 2627 voters in order to be 96 percent confident that the true proportion of U.S. voters opposing capital punishment is estimated to within 0.02.

To determine the sample size needed for estimating a proportion, we can use the formula:

n = (Z^2 * p * q) / E^2

where:

n is the sample size

Z is the z-value corresponding to the desired confidence level (96 percent confidence corresponds to a z-value of approximately 1.75)

p is the estimated proportion (since we don't have an initial estimate, we can assume a conservative estimate of 0.5)

q is 1 - p (complement of the estimated proportion)

E is the desired margin of error (0.02)

Plugging in the values, we have:

n = (1.75^2 * 0.5 * 0.5) / 0.02^2

n ≈ 2627

Therefore, the pollster should survey 2627 voters in order to be 96 percent confident that the true proportion of U.S. voters opposing capital punishment is estimated to within 0.02.

This sample size calculation ensures that the estimate of the true proportion will have a margin of error (E) of 0.02 or less, providing a high level of confidence (96 percent) in the accuracy of the estimate.

In conclusion, by using the formula for sample size estimation, the pollster should survey 2627 voters to achieve a 96 percent confidence level with a margin of error of 0.02 when estimating the true proportion of U.S. voters opposing capital punishment.

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The function can be expressed in terms of natural logarithms as f(x) = (1/2) ln (4x+5) - (1/2) ln (7x-4).

The function is, f(x) = ln √(4x+5)/√(7x-4)

First, note that we can simplify the expression inside the natural logarithm by applying the rules of radicals:

√(4x+5)/√(7x-4) = (4x+5)^(1/2) / (7x-4)^(1/2)

Now, we can apply the rule for the logarithm of a quotient, which states that:

ln (a/b) = ln a - ln b

Using this rule, we can rewrite the given function as:

f(x) = ln [(4x+5)^(1/2) / (7x-4)^(1/2)]

= ln (4x+5)^(1/2) - ln (7x-4)^(1/2)

Next, we can apply the rule for the logarithm of a power, which states that:

ln a^n = n ln a

Using this rule, we can simplify the logarithms in the expression above:

f(x) = (1/2) ln (4x+5) - (1/2) ln (7x-4).

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The function may still be continuous.

The lack of differentiability only implies that the function is not smooth at that point.

Here are three scenarios in which the derivative of a function may not exist, along with a sketch of each situation:

Corner: A function may not be differentiable at a corner, where the graph abruptly changes direction.

At a corner, the left and right-hand limits of the derivative do not match. Here is a sketch of the situation:

                   /

                  /

                 /

                /

               /

---------------/----------------

Cusp: A function may not be differentiable at a cusp, where the graph has a sharp point.

At a cusp, the left and right-hand limits of the derivative approach different values. Here is a sketch of the situation:

           /

          /

         /

        /

       /

------/--------

      \

       \

        \

         \

          \

Vertical tangent: A function may not be differentiable at a vertical tangent, where the graph has a vertical line tangent to the curve.

At a vertical tangent, the derivative is either infinite or undefined. Here is a sketch of the situation:

     |

     |

------|-------

     |

     |

Note that in all three of these situations, the function may still be continuous.

The lack of differentiability only implies that the function is not smooth at that point.

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