Question

on a snow day, caroline created two snowmen in her backyard. snowman a was built to a height of 36 inches and snowman b was built to a height of 57 inches. the next day, the temperature increased and both snowmen began to melt. at sunrise, snowman a’s height decrease by 3 inches per hour and snowman b’s height decreased by 6 inches per hour. let a represent the height of snowman a t hours after sunrise and let b represent the height of snowman b t hours after sunrise. graph each function and determine how tall each snowman is when they are the same height.

Explanation:

Step1: Define the functions

For Snowman A, initial height is 36 inches and decreases by 3 inches per hour. So the function is \( A(t) = 36 - 3t \). For Snowman B, initial height is 57 inches and decreases by 6 inches per hour. So the function is \( B(t) = 57 - 6t \).

Step2: Set the functions equal

To find when they are the same height, set \( A(t)=B(t) \), so \( 36 - 3t = 57 - 6t \).

Step3: Solve for t

Add \( 6t \) to both sides: \( 36 + 3t = 57 \). Subtract 36 from both sides: \( 3t = 21 \). Divide by 3: \( t = 7 \) hours.

Step4: Find the height

Substitute \( t = 7 \) into \( A(t) \): \( A(7)=36 - 3\times7 = 36 - 21 = 15 \) inches. (Or into \( B(t) \): \( 57 - 6\times7 = 57 - 42 = 15 \) inches.)

Answer:

15 inches