Solve each proportion. Show your work.

a . 2x/7 = 12/14

b. 3/x-1 = 5/7

Final answer:

The remainder of 4^31 when divided by 10 is 4, determined by examining the repeating last-digit pattern of powers of 4.

Explanation:

To find the remainder of 4^31 when divided by 10, we can use the concept of powers and properties of numbers related to modulo arithmetic. Specifically, we look at the last digit pattern that emerges when calculating powers of 4, since the remainder when dividing by 10 depends solely on the final digit of the number.

The powers of 4 have a cycle: 4^1 ends with 4, 4^2 ends with 6, 4^3 ends with 4 again, and so on. This cycle repeats every two powers, so we only need to consider the power of 4 modulo 2 to determine the last digit of 4^31. Since 31 is an odd number, we know that 4^31 will end with a 4. Hence, the remainder of 4^31 divided by 10 is 4.

To determine Daniel's rate per lawn mowing, we would need to divide his total earnings by the total number of lawns mowed. The answer can also be found by evaluating the slope of a graph with earnings on the y-axis and lawns mowed on the x-axis as the slope would represent his earnings per mow.

Without the graph provided, we can't determine the exact answer to this problem. However, conceptually, Daniel's rate per lawn would be calculated by dividing the total amount he earned by the total number of lawns he mowed. For example, if he earned $60 from mowing 4 lawns, his rate would be $15 per lawn.

To find the answer using a graph, you would need to evaluate the slope of the line if the y-axis represents earnings and the x-axis represents lawns mowed. The slope of the line provides the rate, which is Daniel's earnings per mow.

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Answer: width = 18.5 ft; area = 342.3 ft2

Step-by-step explanation:

you plug in the width with the function , where ever there is an x replace it with the width for the maximum area

The root of the function is such point [tex]x_0,[/tex] for which [tex]f(x_0)=0.[/tex]

The attached graph of the function  [tex]f(x) = 2x^3 - 19x^2 + 57x - 54[/tex] shows, that the graph of the function intersects the x-axis in three points (2,0), (3,0) and (4.5,0).

This means that for

Thus, x=2, x=3 and x=4.5 are rational roots of the polynomial function f(x).

Answer: three rational roots.

The polynomial function f(x) = 2x^3 - 19x^2 + 57x - 54 has three rational roots at x=2, x=3, and x=4.5, as confirmed by the graph intersecting the x-axis at these points.

The graph of the function f(x) = 2x^3 - 19x^2 + 57x - 54 intersects the x-axis at three points: (2,0), (3,0), and (4.5,0). This implies that x=2, x=3, and x=4.5 are the roots of the polynomial function.

Since these values are rational, the function has three rational roots. Rational roots occur when the function evaluates to zero for specific rational input values.

In this case, f(2) = 0, f(3) = 0, and f(4.5) = 0. The graph crossing the x-axis at these points confirms their status as roots. The polynomial f(x) has three rational roots, which are x = 2, x = 3, and x = 4.5.

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The net change in elevation from the Atacama trench to the Mariana trench is 2.968km.

We know that the deepest part of Mariana trench is at -11.033 km, while the deepest part of the Atacama trench is -8.065km.

To find the net change, we need to find the difference between the largest value and the smallest value.

So we get:

D = -8.065km - (-11.033 km) = 2.968km

Then the correct option is the second one.

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Answer:

      [tex]2x^2+4x+2[/tex]

Step-by-step explanation:

[tex]2x^3+6x^2+6x+2[/tex] divide by x+1

Divide the first term of numerator by the first term of denominator and put it at the top. then multiply the answer by x+1 . write the expression at the bottom and subtract it. Repeat the process till we get remainder.

                                         [tex]2x^2+4x+2[/tex]

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

          [tex]x+1[/tex]            [tex]2x^3+6x^2+6x+2[/tex]

                                [tex]2x^3+2x^2[/tex]

                                     -----------------------------------------(subtract)

                                                   [tex]4x^2+6x[/tex]

                                                    [tex]4x^2+4x[/tex]

                                            -------------------------------------------Subtract

                                                            [tex]2x+2[/tex]

                                                           [tex]2x+2[/tex]

                                             ------------------------------------------Subtract

                                                                 0

To earn $500 in interest at a 4% annual rate, you need to leave $2000 in the account for approximately 6.25 years.

The formula for simple interest is given by:

[tex]\[ I = P \times r \times t \][/tex]

Where:

- I is the interest earned,

- P is the principal amount (initial deposit),

- r is the annual interest rate (expressed as a decimal), and

- t is the time the money is invested or borrowed in years.

In this case, you want to find t, so rearrange the formula to solve for t:

[tex]\[ t = \frac{I}{P \times r} \][/tex]

Given that P = $2000, r = 0.04 (4% as a decimal), and I = $500, plug these values into the formula:

[tex]\[ t = \frac{500}{2000 \times 0.04} \][/tex]

Calculate the result to find out how long you must leave the money in the account to earn $500 in interest.

[tex]\[ t = \frac{500}{2000 \times 0.04} \]\[ t = \frac{500}{80} \]\[ t = 6.25 \][/tex]

Therefore, you must leave the money in the account for approximately 6.25 years to earn $500 in interest at a simple annual rate of 4%.

we have

[tex]f(x)=(x-3)^{2}(x+2)^{2}(x-1)[/tex]

we know that

The zero of the function is  the value of x when the value of the function is equal to zero

[tex]f(x)=(x-3)^{2}(x+2)^{2}(x-1)[/tex]

rewrite the function

[tex]f(x)=(x-3)(x-3)(x+2)(x+2)(x-1)[/tex]

The zero's of the function are

[tex]x=3\ x=3\ x=-2\ x=-2\ x=1[/tex]

so

[tex]x=3[/tex] ------> has a multiplicity of [tex]2[/tex]

[tex]x=-2[/tex] ------> has a multiplicity of [tex]2[/tex]

[tex]x=1[/tex] ------> has a multiplicity of [tex]1[/tex]

therefore

the answer is

The zero [tex]1[/tex] has a multiplicity of [tex]1[/tex]

The zero [tex]-2[/tex] has a multiplicity of [tex]2[/tex]

From the given function, f(x) = (x − 3)2(x + 2)2(x − 1);

The function f(x) = (x − 3)²(x + 2)²(x − 1) given can be expanded so that we have;

Hence, the zeros of the function f(x) are;

We can conclude that the zero 1 has a multiplicity of 1 and the zero -2 has a multiplicity of 2.

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Since Ned's truck suffered body damage due to a hailstorm, the components of Ned's insurance policy will take care of the damage done to Ned's truck during the hailstorm is comprehensive damage.

A comprehensive coverage is an insurance policy which covers the cost of damages to a vehicle which has been involved in an accident that is not caused by a collision.

Comprehensive insurance covers losses such as:

Therefore, the components of Ned's insurance policy will take care of the damage done to Ned's truck during the hailstorm is comprehensive damage.

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Answer:

x+x+x+x = 4x.

Step-by-step explanation:

x + x + x + x can represent 4x since there are 4 x’s. Those 4 x’s represent x being multiplied by 4, since there are 4 x’s.

Therefore, your answer is 4x.

The question revolves around solving a system of three linear equations with three variables, using algebraic techniques such as substitution or elimination.

The system of equations provided can be solved using methods such as substitution, elimination, or matrix operations. The equations given are:

To solve this system, we can perform operations such as eliminating one variable at a time or using matrices to find the values of x, y, and z that satisfy all three equations simultaneously.

The planning value for the population standard deviation can be found by dividing the range by 4.

To estimate the population standard deviation, we can use the planning value formula. The planning value for the population standard deviation is equal to the range divided by 4. Thus, in this case, if the range is estimated to be 36, the planning value for the population standard deviation would be 9.