Question

On a snow day, Hawa created two snowmen in her 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 2 inches per hour and Snowman B's height decreased by 6 inches per hour. Let AA represent the height of Snowman A tt hours after sunrise and let BB represent the height of Snowman B tt hours after sunrise. Write an equation for each situation, in terms of t, commat, and determine the interval of time, t, commat, when Snowman A is taller than Snowman B.

Answer 1

Let's represent the height of each snowman as a function of time \( t \) (in hours after sunrise).

**Snowman A:**
- Initial height: 33 inches
- Decrease rate: 2 inches per hour

The height of Snowman A, \( AA(t) \), can be modeled by the equation:
\[
AA(t) = 33 - 2t
\]

**Snowman B:**
- Initial height: 53 inches
- Decrease rate: 6 inches per hour

The height of Snowman B, \( BB(t) \), can be modeled by the equation:
\[
BB(t) = 53 - 6t
\]

To find the interval of time when Snowman A is taller than Snowman B, we want to solve the inequality:
\[
AA(t) > BB(t)
\]

Substituting in the equations for \( AA(t) \) and \( BB(t) \):
\[
33 - 2t > 53 - 6t
\]

Now, let's solve for \( t \):

1. Add \( 6t \) to both sides:
\[
33 + 4t > 53
\]

2. Subtract 33 from both sides:
\[
4t > 20
\]

3. Divide both sides by 4:
\[
t > 5
\]

Now, we need to check when Snowman A becomes shorter than or equal to Snowman B:

Using the same equations:
Set \( AA(t) \leq BB(t) \):
\[
33 - 2t \leq 53 - 6t
\]

1. Add \( 6t \) to both sides:
\[
33 + 4t \leq 53
\]

2. Subtract 33 from both sides:
\[
4t \leq 20
\]

3. Divide both sides by 4:
\[
t \leq 5
\]

Therefore, combining these results, we find:
- Snowman A is taller than Snowman B when \( t > 5 \) hours.
- Snowman A is shorter than or equal to Snowman B when \( t \leq 5 \) hours.

### Summary:
- The height of Snowman A is greater than that of Snowman B for **\( t > 5 \) hours** after sunrise.