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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.

1 answer

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.