Question

appc lesson 4.4 homework

1. identify whether each scenario represents exponential growth, exponential decay, or neither. give a reason for your answer.

a. the volume of a cube as the side length grows
b. the price of a shirt with a 40% off coupon
c. the reselling value of a car based on its age in years

Explanation:

Part a

Step1: Recall volume formula

The volume \( V \) of a cube with side length \( s \) is \( V = s^3 \).

Step2: Analyze growth type

Exponential growth/decay has the form \( y = ab^x \) (constant base, variable exponent). Here, \( V = s^3 \) is a cubic function (variable base, constant exponent), not exponential. So it's neither.

Step1: Define the price function

Let the original price be \( P_0 \). After a 40% discount, the new price \( P \) is \( P = P_0(1 - 0.40)=P_0(0.60) \). If we consider multiple applications (though here it's one, but the structure is exponential), for \( n \) discounts (or in general, a proportional decrease), it would be \( P = P_0(0.60)^n \), which is exponential decay (base \( 0.6 < 1 \)).

Step2: Confirm decay

Since the price is multiplied by a constant factor (\( 0.6 \)) each time the discount is applied (or in the scenario of repeated discounts, or even as a single - step proportional decrease), it follows the exponential decay model \( y = ab^x \) with \( a = P_0 \), \( b = 0.6<1 \). So it's exponential decay.

Step1: Recall car resale value model

Car resale value typically follows exponential decay, where the value \( V \) of a car with original value \( V_0 \), depreciation rate \( r \), and age \( t \) (in years) is modeled as \( V = V_0(1 - r)^t \). The value decreases by a constant percentage each year, which is the characteristic of exponential decay (base \( 1 - r<1 \) since \( r>0 \) for depreciation).

Step2: Determine the type

Since the reselling value of a car decreases by a proportional amount (percentage) each year as its age increases, it follows the exponential decay model \( y = ab^x \) (with \( a = V_0 \), \( b = 1 - r<1 \), \( x = t \)). So it's exponential decay.

Answer:

Neither, because the volume of a cube (\( V = s^3 \)) is a cubic function, not in the form of exponential growth/decay (\( y = ab^x \)).

Part b