In each problem, a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse. Find each missing length. Round your answer to the nearest tenth.

b if a=8 and c=17

integer or reduced fraction (for ex. , type in as 2/3) in lowest terms. 38/8 * 64/6 */* 6/5, the evaluated expression is 380/9.

To evaluate the expression:

(38/8) * (64/6) / (6/5)

We can simplify each fraction and then perform the multiplication and division:

(38/8) = (19/4)

(64/6) = (32/3)

(6/5) remains the same.

Now we can multiply the fractions:

(19/4) * (32/3) / (6/5)

To multiply fractions, we multiply the numerators together and the denominators together:

(19 * 32) / (4 * 3) / (6/5)

= (608/12) / (6/5)

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

(608/12) * (5/6)

= (608 * 5) / (12 * 6)

= 3040/72

Now we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 8:

(3040/72) / 8/8

= (380/9) / 1

= 380/9

Therefore, the evaluated expression is 380/9.

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The given infinite series Σ[∞]n=1 7(2)ⁿ⁻¹ evaluates to a sum of -7, indicating that as more terms are added, the series converges to -7.

The given series is Σ[∞]n=1 7(2)ⁿ⁻¹.

Identify the first term and common ratio:

The first term, a, is 7.

The common ratio, r, is 2.

Use the formula for the sum of a geometric series

Sum = [tex]a/(1-r)[/tex]

Substitute the values into the formula

Sum = [tex]7/(1-2)[/tex]

Simplify the denominator

Sum = [tex]7/(-1)[/tex]

Divide:

Sum = -7

Therefore, the sum of the infinite series Σ[∞]n=1 7(2)ⁿ⁻¹ is -7.

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We multiply the numerator and denominator by the conjugate of the denominator, which is (3√3 + 2). This gives us the following:

4 / (3√3 - 2) = 4 * (3√3 + 2) / (3√3 - 2)(3√3 + 2) = 12√3 + 8 / 9(3) = 4√3 + 2 / 3

To rationalize a denominator, we multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a number is the number that is obtained by changing the sign of the imaginary part. In this case, the denominator is (3√3 - 2), so the conjugate is (3√3 + 2).

When we multiply the numerator and denominator by the conjugate, we get a new fraction with a simplified denominator. In this case, the simplified denominator is 9(3), which is equal to 27.

We can then simplify the numerator by combining the terms and dividing by the common factor of 2. This gives us the simplified fraction 4√3 + 2 / 3.

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To use the halving and doubling strategy to find 28 ÷ 50, we can express it as (28 ÷ 25) ÷ 2. By halving and doubling both the dividend and divisor, we can simplify the division problem to obtain the result.

The halving and doubling strategy is a technique used to simplify division problems by repeatedly halving the dividend and doubling the divisor until a manageable calculation is reached. To find 28 ÷ 50 using this strategy, we can express it as (28 ÷ 25) ÷ 2.

First, we halve the dividend, 28, by dividing it by 25, resulting in 1.12. Next, we double the divisor, 50, by multiplying it by 2, giving us 100. Therefore, the division problem becomes 1.12 ÷ 100.

This new division problem can be easily solved, resulting in the final answer of 0.0112. By using the halving and doubling strategy, we have simplified the division of 28 ÷ 50 into the division of 1.12 ÷ 100, making it easier to calculate.

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The objective is to maximize the profit, which is determined by the revenues minus costs. The revenues are calculated by multiplying the number of loaves sold by the selling price, which is 30 c (cents) per loaf.

The costs consist of the cost of flour and the opportunity cost of flour (in case there is leftover flour). The constraints are as follows:
Flour Constraint: The amount of flour used in baking each loaf multiplied by the number of loaves baked should not exceed the total amount of flour available (30 oz).
5x ≤ 30

Yeast Constraint: The number of packages of yeast required for each loaf multiplied by the number of loaves baked should not exceed the total number of yeast packages available (5 packages).
1x ≤ 5

Non-negativity Constraint: The number of loaves baked cannot be negative.
x ≥ 0

To maximize the profit, we can formulate the linear programming problem as follows:

Maximize Z = 30x - (4x + 30(30 - 5x)) = 30x - (4x + 900 - 150x)

subject to:
5x ≤ 30
1x ≤ 5
x ≥ 0

Solving this linear programming problem will provide the optimal value for x, representing the number of loaves the baker should bake and sell in order to maximize their profits.

Note: The selling price and cost values used in the objective function are in cents (c), not dollars ($).

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There are a total of 672 possible outcomes for the situation. Each outcome represents a unique combination of size, color, material, and width for the pair of women's shoes.

To determine the number of possible outcomes, we need to consider the different options for each characteristic of the shoes.

For the size, there are 7 whole sizes available (5 through 11).

For the color, there are 4 options (red, navy, brown, black).

For the material, there are 2 options (leather or suede).

For the width, there are 3 different options.

To find the total number of possible outcomes, we multiply the number of options for each characteristic:

7 (sizes) * 4 (colors) * 2 (materials) * 3 (widths) = 672

Therefore, there are a total of 672 possible outcomes for the situation. Each outcome represents a unique combination of size, color, material, and width for the pair of women's shoes.

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Answer:

m = 1/3

Step-by-step explanation:

Given two points on a line, we can find the slope of the line using the slope formula, which is given by:

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

Thus, we can plug in (8, 2) for (x1, y1) and (-7, -3) for (x2, y2) to find m, the slope of the line passing through the two points:

m = (-3 - 2) / (-7 - 8)

m = -5 / -15

m = 1/3

Thus, the slope, m, of the line passing through the points (8, 2) and (-7, -3) is 1/3.

The slope is:

Work/explanation:

Use the slope formula:

[tex]\bf{m=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

where m = slope;

(x₁, y₁) is a point;

(x₂, y₂) is another point.

Label the values:

m is unknown;

(x₁, y₁) is (8,2);

(x₂, y₂) is (-7, -3).

Plug in the data:

[tex]\bf{m=\dfrac{-3-2}{-7-8}}[/tex]

Simplify both the numerator and the denominator

[tex]\bf{m=\dfrac{-5}{-15}}[/tex]

[tex]\bf{m=\dfrac{5}{15}}[/tex]

[tex]\bf{m=\dfrac{1}{3}}[/tex]

Answer:

(a) Vertical asymptote:  x = 5

     Horizontal asymptote:  y = 0

(b) Domain: (-∞, 5) ∪ (5, ∞)

     Range: (-∞, 0)

(c) x-intercept(s): None

     y-intercept: -1

Step-by-step explanation:

A vertical asymptote is a vertical line that the curve gets infinitely close to, but never touches. It is displayed as a vertical dashed line on the given graph. Therefore, the vertical asymptote is:

A horizontal asymptote is a horizontal line that the curve gets infinitely close to, but never touches. It is displayed as a horizontal dashed line on the given graph. Therefore, the horizontal asymptote is:

[tex]\hrulefill[/tex]

Since the graph has a vertical asymptote at x = 5, it means that the function is undefined at x = 5. Therefore, the domain of the graph will be all real numbers except x = 5:

Since there is a horizontal asymptote at y = 0 and the curve appears to be always below the x-axis, it indicates that the range of the graph will be all negative y-values. Therefore, the range of the graphed function is:

[tex]\hrulefill[/tex]

The x-intercepts are the x-values of the points where the curve intersects the x-axis, so when the y-coordinate of a point on the graph is zero.

As the given graph has a horizontal asymptote at y = 0 and the curve appears to be always below the x-axis, it implies that the graph does not cross the x-axis. Therefore:

The y-intercept is the y-value at the point where the curve intersects the y-axis, so when the x-coordinate of a point on the graph is zero.

From inspection of the given graph, we can see that the curve crosses the y-axis at y = -1. Therefore:

The simplified form of the expression [tex]x^{3/2}[/tex](√x - 1/√x) is (√x³) - (√x).

To simplify the given expression, let's break it down step by step. Starting with the term (√x - 1/√x), we can simplify it by finding a common denominator, which in this case is √x. Thus, we have (√x * √x - 1) / √x, which simplifies to (x - 1) / √x.

Next, we have x³/² multiplied by (x - 1) / √x. To simplify the exponent, we can rewrite [tex]x^{3/2}[/tex] as ([tex]x^{3/2}[/tex])), which represents the square root of x cubed. Multiplying this by (x - 1) / √x gives us [([tex]x^{3/2}[/tex])) * (x - 1)] / √x.

To further simplify, we can distribute the exponent of (3/2) to both terms inside the brackets, resulting in [([tex]x^{3/2}[/tex] + 1/2)) * (x - 1)] / √x. Simplifying the exponent gives us [([tex]x^{4/2}[/tex]) * (x - 1)] / √x, which simplifies further to ([tex]x^{2}[/tex] * (x - 1)) / √x.

Finally, we can rewrite [tex]x^{2}[/tex] as √([tex]x^{4}[/tex]) to combine the square root terms. This gives us (√([tex]x^{4}[/tex]) * (x - 1)) / √x, which simplifies to (√[tex]x^{4}[/tex] * (x - 1)) / √x. As √[tex]x^{4}[/tex] is equal to [tex]x^{2}[/tex], the expression simplifies to ([tex]x^{2}[/tex] * (x - 1)) / √x.

In summary, the simplified form of [tex]x^{3/2}[/tex](√x - 1/√x) is (√x³) - (√x), which represents the square root of x cubed minus the square root of x.

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The ninth term in the sequence is 81.

The given sequence 4, 9, 16, 25, 36, ... can be identified as a sequence of perfect squares. The explicit formula for this sequence can be obtained by recognizing that each term is the square of its corresponding natural number position.

The explicit formula for this sequence can be written as:

() = ^2

Where () represents the -th term in the sequence.

To find the ninth term in this sequence, we substitute = 9 into the formula:

(9) = 9^2

= 81

Therefore, the ninth term in the sequence is 81.

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The simplified complex numbers expression is:

(-3 + 2i) - (6 + i) = -9 + i

Given is an expression (-3+2i) - (6+i) containing complex numbers we need to simplify it,

To simplify the expression (-3+2i)-(6+i), we can combine like terms.

First, let's distribute the negative sign to the second parentheses:

(-3+2i) - (6+i) = -3 + 2i - 6 - i

Next, let's combine the real terms (-3 and -6):

(-3 + 2i - 6 - i) = (-3 - 6) + 2i - i

Simplifying the real terms, we have:

(-3 - 6) = -9

Finally, combining the imaginary terms (2i and -i), we get:

2i - i = i

Therefore, the simplified complex numbers expression is:

(-3 + 2i) - (6 + i) = -9 + i

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The given "proof" is incomplete and does no longer provide a convincing argument for the statement that [tex]\sqrt{n}[/tex] is irrational for every natural wide variety of n.

It begins by assuming that [tex]\sqrt{n}[/tex] is rational, represented as [tex]\sqrt{n}[/tex] = a/b, wherein a and b are integers and not using common factors and b isn't equal to zero.

The blunders in this evidence lie within the assumption that [tex]\sqrt{n}[/tex] can be represented as a rational number. The evidence fails to expose a contradiction or offer proof that [tex]\sqrt{n}[/tex] can not be expressed as a ratio of integers. In order to prove that [tex]\sqrt{n}[/tex] is irrational, one has to show that there are not any viable values for a and b that satisfy the equation √n = a/b.

To establish the irrationality of [tex]\sqrt{n}[/tex], legitimate evidence usually utilizes techniques along with evidence with the aid of contradiction or evidence by means of high factorization. These techniques involve assuming that [tex]\sqrt{n}[/tex] is rational, manipulating the equation, and deriving a contradiction or showing that the idea results in a not possible situation.

Since the given proof lacks those crucial elements, it can't establish a declaration that [tex]\sqrt{n}[/tex] is irrational for each natural range n.

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The correct question is:

"What is wrong with the following "proof" of the statement that [tex]\sqrt{n}[/tex] is irrational for every natural number n? "proof ". Suppose that [tex]\sqrt{n}[/tex] is rational is a rational number."

Proof:
1. ∠1 ≅ ∠2 (Given)
2. Let a and b be two lines intersected by a transversal line t

3. Assume, for the sake of contradiction, that a and b are not parallel
4. If a and b are not parallel, then there exists a pair of corresponding angles that are not congruent
5. Let ∠3 be a corresponding angle to ∠1 and ∠4 be a corresponding angle to ∠2
6. By the Corresponding Angles Postulate, if a and b are not parallel, then ∠3 and ∠4 are not congruent

7. However, from statement 1, we know that ∠1 ≅ ∠2
8. Therefore, ∠3 and ∠4 must be congruent as well, contradicting statement 6
9. The assumption made in step 3 is false, so a and b must be parallel
10. Therefore, we have proved that if ∠1 ≅ ∠2, then a || b.

In this proof, we start by assuming that the lines a and b are not parallel. We then show that if ∠1 ≅ ∠2, this assumption leads to a contradiction. By using the Corresponding Angles Postulate and the given information, we establish that ∠3 and ∠4 must be congruent. This contradiction proves that our initial assumption was false, and therefore, a and b must be parallel.

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For the given function with the points (1,2), (2,3), (3,4), (4,5):

Domain = {1, 2, 3, 4}

Range = {2, 3, 4, 5}

To find the domain and range of a function, we need to determine the set of possible input values (domain) and the set of corresponding output values (range).

Given the points: (1,2), (2,3), (3,4), (4,5)

Domain: The set of all input values (x-values) in the given points.

The domain of this function is {1, 2, 3, 4}.

Range: The set of all output values (y-values) in the given points.

The range of this function is {2, 3, 4, 5}.

Therefore, for the given function with the points (1,2), (2,3), (3,4), (4,5):

Domain = {1, 2, 3, 4}

Range = {2, 3, 4, 5}

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The distance from Anica's fist to her shoulder is greater in Position 2 compared to Position 1.

Position 1: In this position, Anica's elbow is fully extended, and her fist is closest to her shoulder. Let's assume the length from Anica's shoulder to her elbow is "a," and the length from her elbow to her fist is "b." In Position 1, the distance from her fist to her shoulder can be calculated as a + b since her elbow is fully extended.

Position 2: In this position, Anica's elbow is flexed, and her fist is further away from her shoulder. Let's assume the new length from her elbow to her fist is "c." In Position 2, the distance from her fist to her shoulder can be calculated as a + c since her elbow is flexed.

Since c is greater than b (as her fist is further away from her shoulder), the distance from Anica's fist to her shoulder is greater in Position 2.

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With an annual dividend of $5.99 expected to grow at a constant rate of 3.5% and a market requirement of a 10.0% return, the stock is worth approximately $91.27.

The dividend discount model is a valuation method that estimates the intrinsic value of a stock by considering the present value of its future dividends. In this case, we can use the DDM formula to calculate the stock's worth:

Stock Price = Dividend / (Required Return - Dividend Growth Rate)

Given that Kraft Heinz pays an annual dividend of $5.99 and the expected growth rate is 3.5%, and the market requires a 10.0% return, we can substitute these values into the formula:

Stock Price = $5.99 / (0.10 - 0.035) = $5.99 / 0.065 ≈ $92.15

Therefore, based on the dividend discount model and the given assumptions, the stock price of Kraft Heinz is approximately $92.15.

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The given production function Y = zK^2N^2 does not satisfy all of the properties of production functions. It violates the property of constant returns to scale.

A production function is a mathematical representation of the relationship between inputs (factors of production) and outputs (goods or services). It is expected to satisfy certain properties for it to be considered a valid production function.

One of the key properties of production functions is constant returns to scale, which means that if all inputs are scaled up or down proportionally, the output should also be scaled up or down by the same factor. In the given production function Y = zK^2N^2, we can observe that the exponents of both capital (K) and labor (N) are 2. This implies that doubling both inputs should result in a quadrupling of output (2^2 * 2^2 = 4). However, this violates the principle of constant returns to scale, as the output is increasing at an increasing rate, not a constant rate.

Therefore, the given production function fails to satisfy the property of constant returns to scale. Other properties such as positive marginal products and non-negativity of inputs may still hold, but without constant returns to scale, the production function does not conform to all the expected properties.

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When r = 0.40 and M = 11, the equation i = (1 + r/M)^(M−1) can be evaluated to find the value of i. Hence i = 0.3479, rounded to four decimal places.

Let's substitute the given values into the equation. We have i = (1 + 0.40/11)^(11−1). Simplifying this, we get i = (1 + 0.0364)^10. Evaluating further, i = 1.0364^10 ≈ 1.3479. By subtracting 1 from this value, we obtain i ≈ 0.3479.

The equation represents the formula for calculating compound interest. In this case, r represents the interest rate, M represents the number of compounding periods in a year, and i represents the effective annual interest rate. By plugging in the given values of r = 0.40 and M = 11, we can determine the value of i. The calculation involves dividing the interest rate by the number of compounding periods in a year, adding 1, and then raising it to the power of the number of compounding periods minus 1. The resulting value, rounded to four decimal places, is approximately 0.3479.

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To find an equation of the plane that is parallel to the xz-plane and located 28 units to the left of the xz-plane, we can consider that the x-coordinate of any point on the plane will be 28 units less than the x-coordinate of any corresponding point on the xz-plane.

In the standard perspective, the equation of the xz-plane is given by x = 0, which means the x-coordinate is always 0.

To create a plane that is parallel to the xz-plane and located 28 units to the left, we need to shift the x-coordinate by subtracting 28.

Therefore, the equation of the plane is x = -28.

This equation indicates that for any point on the plane, the x-coordinate will always be -28, while the y and z coordinates can take any real values.

Note that this equation assumes a standard coordinate system where the x-axis is horizontal, the y-axis is vertical, and the z-axis is perpendicular to the xz-plane.

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The 18th term of the arithmetic sequence is 45.2, while the balance in an account with a $8,600 deposit and 12% annual interest after 5 years is $14,311.39. With a $6,284 deposit and 14% annual interest after 30 months, the account balance will be $7,463.17. Borrowing $1,709 at a 182% annual interest rate, the monthly payment for 16 months will be $202.06.

1. Arithmetic sequence: The formula to find the nth term of an arithmetic sequence is given by:

nth term = first term + (n - 1) * common difference

Here, the 4th term is given as 6 and the common difference is 2.9.   Plugging in these values, we can calculate the 18th term as follows:

18th term = 6 + (18 - 1) * 2.9 = 6 + 17 * 2.9 = 45.2

2. Compound interest: For the first scenario, where $8,600 is deposited into an account that pays 12% simple annual interest compounded monthly for 5 years, we can calculate the final balance using the formula for compound interest:

A = P * [tex](1 + r/n)^{(n*t) }[/tex]

Here, P is the principal amount, r is the annual interest rate (in decimal form), n is the number of times interest is compounded per year, and t is the number of years. Plugging in the values:

P = $8,600, r = 12% = 0.12, n = 12 (monthly compounding), t = 5

A = 8600 * [tex](1 + 0.12/12)^{(12*5)}[/tex]= $14,311.39

3. Similarly, for the second scenario, where $6,284 is deposited into an account that pays 14% simple annual interest compounded monthly for 30 months:

P = $6,284, r = 14% = 0.14, n = 12 (monthly compounding), t = 30/12 = 2.5

A = 6284 * [tex](1 + 0.14/12)^{(12*2.5)}[/tex] = $7,463.17

4. Monthly payment: To calculate the monthly payment amount for borrowing $1,709 from The Saver at a 182% simple annual interest rate, we can divide the total amount by the number of payments. The formula for calculating the monthly payment for a loan is:

Monthly payment = Total amount / Number of payments

Here, the total amount is $1,709 and the number of payments is 16. Plugging in the values:

Monthly payment = 1709 / 16 = $202.06

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The intention is to bridge the achievement gap and provide equal educational opportunities for all students, regardless of their socioeconomic background.

In the teacher school leader incentive program, there is a requirement that over half of the schools in a funded district have to have over 50% of students eligible for Free or Reduced Price Lunch (FRPL). This requirement aims to address and support schools with a significant proportion of economically disadvantaged students.

The purpose of this requirement is likely to target districts where a substantial number of students come from low-income backgrounds. By focusing on districts with a high percentage of FRPL-eligible students, the program aims to provide additional resources and incentives to improve educational outcomes for these disadvantaged students.

Having over half of the schools meet this criterion ensures that a majority of schools in the district, which are likely to serve a significant portion of the student population, receive additional support. This can include financial resources, professional development opportunities, and other incentives to attract and retain effective teachers and school leaders.

By targeting districts with higher levels of economic disadvantage, the program aims to promote equity in education by providing additional resources to schools serving students who may face unique challenges associated with poverty. The intention is to bridge the achievement gap and provide equal educational opportunities for all students, regardless of their socioeconomic background.

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To find the smallest and largest values of x when the function[tex]f(x) = (3/2)x^3 - 2x^2 - 10x + 8[/tex] has a slope of 0, we need to determine the critical points of the function. The smallest value of x corresponds to the local minimum point, while the largest value of x corresponds to the local maximum point.

To find the critical points, we need to calculate the derivative of the function f(x) with respect to x and set it equal to 0. Taking the derivative of f(x), we get [tex]f'(x) = 4.5x^2 - 4x - 10.[/tex] To find the critical points, we set f'(x) = 0 and solve for x.

Setting [tex]4.5x^2 - 4x - 10 = 0[/tex], we can use the quadratic formula or factoring to find the values of x. However, in this case, the quadratic equation does not factor easily, so we can use the quadratic formula: [tex]x = (-b ± √(b^2 - 4ac)) / 2a[/tex]. Substituting the values a = 4.5, b = -4, and c = -10 into the quadratic formula, we can find the two values of x corresponding to the critical points.

Once we have the critical points, we can determine which one represents the local minimum (the smallest x value) and which one represents the local maximum (the largest x value). By analyzing the concavity of the function or evaluating the second derivative, we can determine which critical point is the minimum and which one is the maximum.

In this case, the smallest value of x corresponds to the local minimum, while the largest value of x corresponds to the local maximum.

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The measure of the angle between the diagonal and the top right portion of the second track is 120 degrees.

Let us assume the angle between the bottom right portion and diagonal of the first track be x. Thus, the another angle, which is the angle between top right portion of the second track and diagonal will be 2x. Now, these two angles are related to each other as consecutive interior angles.

Thus, they are mathematically related to each other as -

x + 2x = 180

Adding the values

3x = 180

Rearranging the equation

x = 180/3

Dividing the values

x = 60

The angle between diagonal and top right portion is 2x. So, angle = 2 × 60

Angle = 120 degrees.

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The volume of sphere : V = 4188.790ft³

The volume of hemisphere: V =  2094.39 ft³.

Given,

Radius = 10ft.

Now,

The Volume of sphere = 4/3 ×π×r³

Substitute the value of r to get the volume,

V = 4/3 ×π × 10³

V = 4188.790ft³

Now ,

Volume of hemisphere = 2/3 × π ×r³

Volume of hemisphere = 2/3 × π × 10³

Volume of hemisphere = 2094.39 ft³.

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It will take approximately 7.27 years for your mother's investment to become worth $1,000,000 when invested in a well-diversified, low-cost mutual fund returning 10% per year.

To determine the number of years it will take for your mother's investment to become worth $1,000,000, we can use the future value formula:

FV = PV * (1 + r)^n

Where:

FV = Future value ($1,000,000)

PV = Present value ($500,000)

r = Annual interest rate (10% or 0.10)

n = Number of years (unknown)

Substituting the given values into the formula:

$1,000,000 = $500,000 * (1 + 0.10)^n

Simplifying the equation:

2 = (1.10)^n

Taking the logarithm of both sides:

log(2) = log(1.10)^n

Using the logarithmic property:

log(2) = n * log(1.10)

Solving for n:

n = log(2) / log(1.10)

Using a calculator:

n ≈ 7.27

Therefore, it will take approximately 7.27 years for your mother's investment to become worth $1,000,000 when invested in a well-diversified, low-cost mutual fund returning 10% per year.

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The answer is:

Work/explanation:

We're asked to solve the equation for L.

So first, I will subtract 10c on each side:

[tex]\sf{20L+10C=1200[/tex]

[tex]\sf{20L=1200-10C}[/tex]

Now, divide each term by 20:

[tex]\sf{L=60-\dfrac{10}{20}C}[/tex]

Simplify

[tex]\sf{L=60-\dfrac{1}{2}C}[/tex]

Answer:   L  =  60  -  1/2C

Therefore, The Solution to the Original Equation is:

L  =  60  -  1/2C

        Isolate the  L Terms:

        20L  =  1200   -   10C

        L  =  1200  -  10C / 20

        L   =  1200 / 20   -   10C / 20

        L   =   60   -   1/2C

       Therefore, The Solution to the Original Equation is:

        L  =  60  -  1/2C

I hope this helps you!

The quadratic function f(x) = x² - 4x - 1 has a vertex, axis of symmetry, and intercepts. The vertex is located at (2, -5), and the axis of symmetry is x = 2. The function intersects the x-axis at approximately (-0.24, 0) and (4.24, 0), and it intersects the y-axis at (0, -1).

To find the vertex of the quadratic function f(x) = x² - 4x - 1, we first need to determine the x-coordinate of the vertex. The formula for the x-coordinate of the vertex of a quadratic function in the form f(x) = ax² + bx + c is given by x = -b / (2a). In this case, a = 1 and b = -4, so the x-coordinate of the vertex is x = -(-4) / (2 * 1) = 4 / 2 = 2.

To find the corresponding y-coordinate of the vertex, we substitute the x-coordinate back into the function. f(2) = (2)² - 4(2) - 1 = 4 - 8 - 1 = -5. Therefore, the vertex is located at (2, -5).

The axis of symmetry is a vertical line that passes through the vertex. Since the x-coordinate of the vertex is 2, the axis of symmetry is x = 2.

To find the x-intercepts of the function, we set f(x) = 0 and solve for x. In this case, we have x² - 4x - 1 = 0. Using the quadratic formula, x = (-(-4) ± √((-4)² - 4(1)(-1))) / (2(1)). Simplifying this expression gives x = (4 ± √(16 + 4)) / 2, which further simplifies to x = (4 ± √20) / 2. Therefore, the x-intercepts are approximately (-0.24, 0) and (4.24, 0).

To find the y-intercept, we substitute x = 0 into the function. f(0) = (0)² - 4(0) - 1 = -1. Therefore, the y-intercept is (0, -1).

In summary, the quadratic function f(x) = x² - 4x - 1 has a vertex at (2, -5), an axis of symmetry at x = 2, x-intercepts at approximately (-0.24, 0) and (4.24, 0), and a y-intercept at (0, -1).

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The probability of Elena being chosen is 60%.

We have,

The concept used in determining the probability of Elena being chosen as the goalkeeper is the complement rule.

The complement rule states that the probability of an event occurring is equal to 1 minus the probability of the event not occurring.

If the chance of Anna being chosen for the women's hockey team is 40%, it means that the probability of Elena being chosen as the goalkeeper is 60%

(since they are the only goalkeepers available for selection, and the probabilities must add up to 100%).

Therefore,

The probability of Elena being chosen is 60%.

Learn more about probability here:

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