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Chapter 5: Problem 70

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of \(p\) dollars each is given by the polynomial \(R(p)=-4 p^{2}+420 p\). Find the revenue received when \(p=90\) dollars.

Short Answer

Expert verified
The revenue is \(5400 when \)p = 90$.

Step by step solution

01

Identify the given polynomial

The revenue polynomial provided is: \[ R(p) = -4p^2 + 420p \].
02

Substitute the given price into the polynomial

Plug in the given value of \(p = 90\) into the polynomial. So, \[ R(90) = -4(90)^2 + 420(90). \]
03

Calculate the squared term

First compute \((90)^2\): \[ (90)^2 = 8100. \]
04

Multiply the squared term by -4

Now multiply 8100 by -4: \[ -4 \times 8100 = -32400. \]
05

Calculate the linear term

Next, multiply the linear term, 420, by 90: \[ 420 \times 90 = 37800. \]
06

Add the results of the terms

Combine the results from the previous steps: \[ R(90) = -32400 + 37800 = 5400. \]
07

Write the final result

The revenue received when \(p = 90\) dollars is \[ R(90) = 5400. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Evaluation
Evaluating a polynomial means finding the value of the polynomial for a given value of its variable. In this exercise, we have the polynomial 1. Start by identifying the polynomial provided:
Substitution Method
The substitution method involves replacing the variable in the polynomial with a given numeric value. For our problem, the variable is the price ( To use this method: Step 2: Substitute
Revenue Function
The revenue function represents the amount of money earned from selling a certain quantity of products at a specified price. In this case, we are given the revenue function The goal is to determine how much revenue the manufacturer earns when shoes are sold at $90 each. The steps to calculate revenue are:

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