On a snow day, Matthew created two snowmen in his backyard. Snowman A was built to a
height of 33 inches and Snowman B was built to a height of 53 inches. The next day, the
temperature increased and both snowmen began to melt. At sunrise, Snowman A’s height
decrease by 5 inches per hour and Snowman B’s height decreased by 10 inches per hour. Let
represent the height of Snowman A hours after sunrise and let represent the height of
Snowman B hours after sunrise. Write an equation for each situation, in terms of , and
determine the interval of time, , when Snowman A is taller than Snowman B.

Ask by Marshall Floyd. in the United States

Mar 06,2025

Question

On a snow day, Matthew created two snowmen in his backyard. Snowman A was built to a
height of 33 inches and Snowman B was built to a height of 53 inches. The next day, the
temperature increased and both snowmen began to melt. At sunrise, Snowman A’s height
decrease by 5 inches per hour and Snowman B’s height decreased by 10 inches per hour. Let
represent the height of Snowman A hours after sunrise and let represent the height of
Snowman B hours after sunrise. Write an equation for each situation, in terms of , and
determine the interval of time, , when Snowman A is taller than Snowman B.

Ask by Marshall Floyd. in the United States

Mar 06,2025

UpStudy AI · Accepted Answer

To solve the problem, we need to establish equations for the heights of Snowman A and Snowman B as they melt over time. ### Step 1: Define the equations 1. **Height of Snowman A**: - Initial height: 33 inches - Decrease rate: 5 inches per hour - Equation: $$ A(t) = 33 - 5t $$ 2. **Height of Snowman B**: - Initial height: 53 inches - Decrease rate: 10 inches per hour - Equation: $$ B(t) = 53 - 10t $$ ### Step 2: Set up the inequality We want to find the interval of time $$ t $$ when Snowman A is taller than Snowman B. This can be expressed as: $$ A(t) > B(t) $$ Substituting the equations we derived: $$ 33 - 5t > 53 - 10t $$ ### Step 3: Solve the inequality Now, we will solve the inequality step by step. 1. Rearranging the inequality: $$ 33 - 5t > 53 - 10t $$ Add $$ 10t $$ to both sides: $$ 33 + 5t > 53 $$ Subtract 33 from both sides: $$ 5t > 20 $$ Divide by 5: $$ t > 4 $$ ### Step 4: Determine the interval Now we need to consider the context of the problem. Since both snowmen are melting, we also need to find when they will reach a height of 0 inches. 1. **For Snowman A**: $$ 33 - 5t = 0 \implies 5t = 33 \implies t = \frac{33}{5} = 6.6 \text{ hours} $$ 2. **For Snowman B**: $$ 53 - 10t = 0 \implies 10t = 53 \implies t = \frac{53}{10} = 5.3 \text{ hours} $$ ### Step 5: Combine the intervals Snowman A is taller than Snowman B when $$ t > 4 $$ hours, but we also need to consider when both snowmen are still standing. - Snowman A will be standing until $$ t = 6.6 $$ hours. - Snowman B will be standing until $$ t = 5.3 $$ hours. Thus, the interval when Snowman A is taller than Snowman B is: $$ 4 < t < 5.3 $$ ### Final Answer The interval of time $$ t $$ when Snowman A is taller than Snowman B is: $$ (4, 5.3) $$ Snowman A is taller than Snowman B between 4 and 5.3 hours after sunrise.

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