Math Problem Statement

find the average rate of change if N(p) = 90- 3psquared find the average rate of change from 3 to 4 dollars and N(p) is boxes of nails at p dollars per box

Solution

The average rate of change of a function N(p)N(p) over an interval [a,b][a, b] is given by the formula:

Average Rate of Change=N(b)N(a)ba\text{Average Rate of Change} = \frac{N(b) - N(a)}{b - a}

Given function:

N(p)=903p2N(p) = 90 - 3p^2

Interval: p=3p = 3 to p=4p = 4

Step 1: Compute N(3)N(3)

N(3)=903(3)2=903(9)=9027=63N(3) = 90 - 3(3)^2 = 90 - 3(9) = 90 - 27 = 63

Step 2: Compute N(4)N(4)

N(4)=903(4)2=903(16)=9048=42N(4) = 90 - 3(4)^2 = 90 - 3(16) = 90 - 48 = 42

Step 3: Compute the average rate of change

N(4)N(3)43=426343=211=21\frac{N(4) - N(3)}{4 - 3} = \frac{42 - 63}{4 - 3} = \frac{-21}{1} = -21

Final Answer:

21 boxes per dollar\mathbf{-21} \text{ boxes per dollar}

This means that as the price increases from $3 to $4, the number of boxes of nails sold decreases by 21 on average per dollar increase.


Do you need a more detailed explanation or have any questions?

Here are some related questions:

  1. What does a negative average rate of change indicate?
  2. How would the rate of change differ if we used a different interval?
  3. What is the instantaneous rate of change at p=3p = 3?
  4. How does the function N(p)N(p) behave for large values of pp?
  5. What happens if the price keeps increasing—will N(p)N(p) ever be negative?

Tip: The average rate of change is the slope of the secant line connecting two points on a function's graph.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rate of Change
Quadratic Functions

Formulas

Average Rate of Change = (N(b) - N(a)) / (b - a)

Theorems

Slope of the Secant Line

Suitable Grade Level

Grades 9-12