5.8 Modeling Using Variation

The quantity $y$ varies directly with the cube of $x .$ If $y = 25$ when $x = 2 ,$ find $y$ when $x$ is 6.

The quantity $y$ varies directly with the square of $x .$ If $y = 24$ when $x = 3 ,$ find $y$ when $x$ is 4.

A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.

A quantity $y$ varies inversely with the cube of $x .$ If $y = 25$ when $x = 2 ,$ find $y$ when $x$ is 6.

A quantity $y$ varies inversely with the square of $x .$ If $y = 8$ when $x = 3 ,$ find $y$ when $x$ is 4.

A quantity $x$ varies directly with the square of $y$ and inversely with the cube root of $z .$ If $x = 6$ when $y = 2$ and $z = 8 ,$ find $x$ when $y = 1$ and $z = 27.$

A quantity $x$ varies directly with the square of $y$ and inversely with $z .$ If $x = 40$ when $y = 4$ and $z = 2 ,$ find $x$ when $y = 10$ and $z = 25.$

What is true of the appearance of graphs that reflect a direct variation between two variables?

Is there a limit to the number of variables that can vary jointly? Explain.

Write an equation describing the relationship of the given variables.

$y$ varies directly as the square of $x$ and when $x = 4 , y = 80 .\textrm{ }$

Write an equation describing the relationship of the given variables.

$y$ varies directly as the cube of $x$ and when $x = 36 , y = 24.$

Write an equation describing the relationship of the given variables.

$y$ varies directly as the fourth power of $x$ and when $x = 1 , y = 6.$

Write an equation describing the relationship of the given variables.

$y$ varies inversely as the square of $x$ and when $x = 3 , y = 2.$

Write an equation describing the relationship of the given variables.

$y$ varies inversely as the fourth power of $x$ and when $x = 3 , y = 1.$

Write an equation describing the relationship of the given variables.

$y$ varies inversely as the cube root of $x$ and when $x = 64 , y = 5.$

Write an equation describing the relationship of the given variables.

$y$ varies jointly as $x , z ,$ and $w$ and when $x = 1 , z = 2 , w = 5 ,$ then $y = 100.$

Write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and the square root of $z$ and when $x = 2$ and $z = 25 ,$ then $y = 100.$

Write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and $z$ and inversely as $w$. When $x = 3 , z = 5$, and $w = 6$, then $y = 10.$

Write an equation describing the relationship of the given variables.

$y$ varies jointly as $x$ and $z$ and inversely as the square root of $w$ and the square of $t .$ When $x = 3 , z = 1 , w = 25 ,$ and $t = 2 ,$ then $y = 6.$

Use the given information to find the unknown value.

$y$ varies directly as the square of $x .$ When $x = 2 ,$ then $y = 16.$ Find $y$ when $x = 8.$

Use the given information to find the unknown value.

$y$ varies directly as the square root of $x .$ When $x = 16 ,$ then $y = 4.$ Find $y$ when $x = 36.$

Use the given information to find the unknown value.

$y$ varies inversely with $x .$ When $x = 3 ,$ then $y = 2.$ Find $y$ when $x = 1.$

Use the given information to find the unknown value.

$y$ varies inversely with the cube of $x .$ When $x = 3 ,$ then $y = 1.$ Find $y$ when $x = 1.$

Use the given information to find the unknown value.

$y$ varies inversely with the cube root of $x .$ When $x = 27 ,$ then $y = 5.$ Find $y$ when $x = 125.$

Use the given information to find the unknown value.

$y$ varies jointly as $x , z , \text{and} w .$ When $x = 2 ,$ $z = 1 ,$ and $w = 12 ,$ then $y = 72.$ Find $y$ when $x = 1 ,$ $z = 2 ,$ and $w = 3.$

Use the given information to find the unknown value.

$y$ varies jointly as the square of $x$ and the square root of $z .$ When $x = 2$ and $z = 9 ,$ then $y = 24.$ Find $y$ when $x = 3$ and $z = 25.$

Use the given information to find the unknown value.

$y$ varies jointly as the square of $x$ and the cube of $z$ and inversely as the square root of $w .\textrm{ }$ When $x = 2 ,$ $z = 2 ,$ and $w = 64 ,$ then $y = 12.$ Find $y$ when $x = 1 ,$ $z = 3 ,$ and $w = 4.$

Use Kepler’s Law, which states that the square of the time, $T ,$ required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a ,$ that the planet is from the Sun.

Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

Use Kepler’s Law, which states that the square of the time, $T ,$ required for a planet to orbit the Sun varies directly with the cube of the mean distance, $a ,$ that the planet is from the Sun.

Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

Use the given information to answer the questions.

The distance $s$ that an object falls varies directly with the square of the time, $t ,$ of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

Use the given information to answer the questions.

The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

Use the given information to answer the questions.

The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

Use the given information to answer the questions.

The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

Use the given information to answer the questions.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

Find the equation of the quadratic function using the given information.

The vertex is $\left(– 2 , 3\right)$ and a point on the graph is $\left(3 , 6\right) .$

Complete the task.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

Determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

$f \left(x\right) = 4 x^{5} - 3 x^{3} + 2 x - 1$

Determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

$f \left(x\right) = x^{2} \left(3 - 6 x + x^{2}\right)$

Determine end behavior of the polynomial function.

$f \left(x\right) = 4 x^{3} - 6 x^{2} + 2$

Find all zeros of the polynomial function, noting multiplicities.

$f \left(x\right) = \left(x + 3\right)^{2} \left(2 x - 1\right) \left(x + 1\right)^{3}$

Find all zeros of the polynomial function, noting multiplicities.

$f \left(x\right) = x^{3} - 4 x^{2} + x - 4$

Use long division to find the quotient and remainder.

$\frac{x^{3} - 2 x^{2} + 4 x + 4}{x - 2}$

Use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

$\frac{x^{3} - 2 x^{2} + 5 x - 1}{x + 3}$

Use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

$\frac{2 x^{3} + 6 x^{2} - 11 x - 12}{x + 4}$

Use the Rational Zero Theorem to help you solve the polynomial equation.

$2 x^{3} - 3 x^{2} - 18 x - 8 = 0$

Use the Rational Zero Theorem to help you solve the polynomial equation.

$2 x^{4} - 17 x^{3} + 46 x^{2} - 43 x + 12 = 0$

Use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

$x^{3} - 3 x^{2} - 2 x + 4 = 0$

Find the slant asymptote.

$f \left(x\right) = \frac{x^{2} - 1}{x + 2}$

Find the inverse of the function with the domain given.

$f \left(x\right) = \left(x - 2\right)^{2} , x \geq 2$

Find the inverse of the function with the domain given.

$f \left(x\right) = x^{2} + 6 x - 2 , x \geq - 3$

Find the inverse of the function with the domain given.

$f \left(x\right) = \sqrt{4 x + 5} - 3$

Find the unknown value.

$y$ varies directly as the square of $x .$ If when $x = 3 , y = 36 ,$ find $y$ if $x = 4.$

Find the unknown value.

$y$ varies jointly as the cube of $x$ and as $z .$ If when $x = 1$ and $z = 2 ,$ $y = 6 ,$ find $y$ if $x = 2$ and $z = 3.$

Solve the application problem.

The weight of an object above the surface of the earth varies inversely with the square of the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

Give the degree and leading coefficient of the following polynomial function.

$f \left(x\right) = x^{3} \left(3 - 6 x - 2 x^{2}\right)$

Determine the end behavior of the polynomial function.

$f \left(x\right) = - 2 x^{2} \left(4 - 3 x - 5 x^{2}\right)$

Given information about the graph of a quadratic function, find its equation.

Vertex $\left(2 , 0\right)$ and point on graph $\left(4 , 12\right) .$

Find all zeros of the following polynomial functions, noting multiplicities.

$f \left(x\right) = \left(x - 3\right)^{3} \left(3 x - 1\right) \left(x - 1\right)^{2}$

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

$\frac{x^{4} + 3 x^{2} - 4}{x - 2}$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f \left(x\right) = 2 x^{3} + 5 x^{2} - 6 x - 9$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f \left(x\right) = 4 x^{4} + 16 x^{3} + 13 x^{2} - 15 x - 18$

Given the following information about a polynomial function, find the function.

It has a double zero at $x = 3$ and zeros at $x = 1$ and $x = - 2$ . Its y-intercept is $\left(0 , 12\right) .$

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

$8 x^{3} - 21 x^{2} + 6 = 0$

Find the inverse of the function.

$f \left(x\right) = \sqrt{x - 2} + 4$

Find the inverse of the function.

$f \left(x\right) = \frac{2 x + 3}{3 x - 1}$

Find the unknown value.

$y$ varies jointly with $x$ and the cube root of $z .$ If when $x = 2$ and $z = 27 ,$ $y = 12 ,$ find $y$ if $x = 5$ and $z = 8.$

Perform the indicated operation with complex numbers.

$\left(4 + 3 i\right) + \left(- 2 - 5 i\right)$

Perform the indicated operation with complex numbers.

$\left(2 - 3 i\right) \left(3 + 6 i\right)$

Solve the following equations over the complex number system.

$x^{2} - 4 x + 5 = 0$

Perform the indicated operation or solve the equation.

$\left(3 - 4 i\right) \left(4 + 2 i\right)$

Perform the indicated operation or solve the equation.

$x^{2} - 4 x + 13 = 0$

Determine the end behavior of the polynomial function.

$f \left(x\right) = 8 x^{3} - 3 x^{2} + 2 x - 4$

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

Find all zeros of the following polynomial functions, noting multiplicities.

$f \left(x\right) = 2 x^{6} - 12 x^{5} + 18 x^{4}$

Use long division to find the quotient.

$\frac{2 x^{3} + 3 x - 4}{x + 2}$

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

$\frac{2 x^{3} + 5 x^{2} - 7 x - 12}{x + 3}$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f \left(x\right) = 4 x^{4} + 8 x^{3} + 21 x^{2} + 17 x + 4$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f \left(x\right) = x^{5} + 6 x^{4} + 13 x^{3} + 14 x^{2} + 12 x + 8$

Given the following information about a polynomial function, find the function.

It has a zero of multiplicity 3 at $x = \frac{1}{2}$ and another zero at $x = - 3$. It contains the point $\left(1 , 8\right) .$

Find the slant asymptote of the rational function.

$f \left(x\right) = \frac{x^{2} + 3 x - 3}{x - 1}$

Find the inverse of the function.

$f \left(x\right) = 3 x^{3} - 4$

Find the unknown value.

$y$ varies inversely as the square of $x$ and when $x = 3 ,$ $y = 2.$ Find $y$ if $x = 1.$

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?