Question 498·Medium·Math

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A collector bought an antique vase on January 1. The function $p(t)=120(1.07)^t$ , where $0\le t\le 6$ , gives the predicted price, in dollars, of the vase $t$ years after the purchase.

What is the best interpretation of the statement " $p(5)$ is approximately equal to $168$ " in this context?

A function value pairs an input with the output the rule gives. In a word problem, identify what each one measures, then evaluate the function in Desmos if you need to check the number. The input tells you when; the output tells you the amount. A growth percentage describes a change between two times, not the meaning of one function value.

Hints

Hint 1
In function notation, the number in parentheses is the input. The prompt measures the input in years after purchase. What time does an input of $5$ describe?
Hint 2
The function gives the vase’s predicted price in dollars, not the dollars gained since purchase. When you evaluate it at $t=5$ , does the output describe a total price or a change?

Step-by-step

Read the input, then evaluate
Step 1
Identify the time
The $5$ in $p(5)$ is the input, or the value of $t$ . Since $t$ measures years after purchase, $p(5)$ refers to five years after the vase was bought. That time is allowed because $0\le t\le6$ .
Step 2
Read the price at that time
Type $p(t)=120(1.07)^t$ , then $p(5)$ into Desmos. It shows $p(5)\approx168.31$ . The function’s output is a predicted price in dollars, so this is about $168 at that time, not $168 gained since purchase. The $1.07$ describes year-to-year growth, not what $p(5)$ itself means. Five years after purchase, the vase’s predicted price is approximately $168. Choice D.

Question 498·Medium·Math

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