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 \( t \) hours after sunrise and let \( B \) represent the height of Snowman B \( t \) hours after sunrise. Write an equation for each situation, in terms of \( t \), and determine the interval of time, \( t \), when Snowman A is taller than Snowman B.

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: \[ (4, 5.3) \]