Question

when you own a pet, there are one-time costs, such as a leash, and continuing costs, such as food. suppose the one-time costs for owning a small dog are $180, and the continuing costs are $400 per year.

which function models the average annual cost of owning a small dog for x years?

the average annual cost of owning a small dog for 10 years is $

the function $g(x) = \frac{140 + 500x}{x}$ models the average annual cost of owning a cat for x years. on average, which pet costs more per year over a 10-year period?

on average, how much more is the total cost of owning a cat for 10 years than owning a small dog for 10 years? $

Explanation:

Step1: Find the function for dog's average cost

Total cost for \( x \) years: one - time cost (\( \$180 \)) + continuing cost (\( 400x \)). Average annual cost \( f(x)=\frac{180 + 400x}{x} \).

Step2: Average cost for dog in 10 years

Substitute \( x = 10 \) into \( f(x)=\frac{180+400x}{x} \).
\( f(10)=\frac{180 + 400\times10}{10}=\frac{180 + 4000}{10}=\frac{4180}{10}=418 \).

Step3: Average cost for cat in 10 years

For cat, \( g(x)=\frac{140 + 500x}{x} \). Substitute \( x = 10 \):
\( g(10)=\frac{140+500\times10}{10}=\frac{140 + 5000}{10}=\frac{5140}{10}=514 \).
Compare \( f(10) = 418 \) and \( g(10)=514 \). So cat costs more.

Step4: Total cost difference for 10 years

Total cost for dog in 10 years: \( 180+400\times10 = 4180 \).
Total cost for cat in 10 years: \( 140 + 500\times10=5140 \).
Difference: \( 5140 - 4180=960 \).

Answer:

1. The function is \( \boldsymbol{f(x)=\frac{180 + 400x}{x}} \)
2. The average annual cost for dog in 10 years is \( \boldsymbol{418} \)
3. The pet that costs more is \( \boldsymbol{\text{cat}} \)
4. The total cost difference is \( \boldsymbol{960} \)