A company’s profit $ P $ in thousands of dollars is modeled by the quadratic function $ P(x) = -2x^2 + 12x - 16 $, where $ x $ is the number of units sold in thousands. Determine the number of units sold that will maximize the profit. - 1.3 Rates of Change in Linear and Quadratic Functions - Review - Use this assignment in your class!

1.3 Rates of Change in Linear and Quadratic Functions - Review

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A water tank is being filled at a constant rate. After 2 hours, the tank contains 40 gallons of water, and after 5 hours, it contains 100 gallons. Calculate the constant rate of change and interpret its meaning.

Explain how the vertex form of a quadratic function $f (x) = a (x - h)^{2} + k$ helps in determining the rate of change and the shape of the graph.

Describe how the second derivative of a quadratic function $f (x) = a x^{2} + b x + c$ relates to the concavity and rate of change of the function.

How does the discriminant of a quadratic function $a x^{2} + b x + c = 0$ affect the shape and the rate of change of its graph?

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10.

How does the axis of symmetry of a quadratic function $f (x) = a x^{2} + b x + c$ relate to the rate of change on either side of the vertex?

11.

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Write a linear function that models the total cost $C$ of renting a car and driving it $m$ miles. Then, calculate the total cost if a customer drives 150 miles.

12.

A ball is thrown into the air with an initial velocity of 20 m/s from a height of 2 meters. The height $h$ of the ball at any time $t$ seconds can be modeled by the quadratic function $h (t) = - 5 t^{2} + 20 t + 2$ . Determine the maximum height reached by the ball.

13.

A company’s profit $P$ in thousands of dollars is modeled by the quadratic function $P (x) = - 2 x^{2} + 12 x - 16$ , where $x$ is the number of units sold in thousands. Determine the number of units sold that will maximize the profit.

14.

A water tank is being filled at a constant rate. The volume $V$ of water in the tank after $t$ minutes is given by the linear function $V (t) = 5 t + 10$ . How long will it take to fill the tank to a volume of 60 liters?

15.

A farmer wants to create a rectangular pen with an area of 200 square meters using a quadratic function. If the length of the pen is $x$ meters, express the width $w$ as a function of $x$ and then find the dimensions of the pen that will minimize the perimeter.

A company’s profit $P$ in thousands of dollars is modeled by the quadratic function $P (x) = - 2 x^{2} + 12 x - 16$ , where $x$ is the number of units sold in thousands. Determine the number of units sold that will maximize the profit.

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AI InstructorAUG 11, 2024, 3:04:34 PM

To find the number of units sold that will maximize the profit, we need to find the vertex of the quadratic function $P (x) = - 2 x^{2} + 12 x - 16$ . The number of units $x$ at which the maximum profit occurs is given by $x = - \frac{b}{2 a}$ , where $a = - 2$ and $b = 12$ :
$x = - \frac{12}{2 \times - 2} = 3$
So, the number of units sold that will maximize the profit is 3 thousand units.

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A company’s profit P in thousands of dollars is modeled by the quadratic function P(x)=−2x2+12x−16, where x is the number of units sold in thousands. Determine the number of units sold that will maximize the profit.

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A company’s profit $P$ in thousands of dollars is modeled by the quadratic function $P (x) = - 2 x^{2} + 12 x - 16$ , where $x$ is the number of units sold in thousands. Determine the number of units sold that will maximize the profit.