Question

benito and tyler are painting opposite sides of the same fence. tyler has already painted 19\frac{1}{2} feet of his side of the fence when benito starts painting. answer parts a to c.\
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benito: painting rate 15 ft/min\
tyler: painting rate 3 ft/min\
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\textbf{a.} how long will it take for the two sides of the fence to have an equal number of feet painted? how many feet will be painted on benito’s side of the fence when he catches up with tyler? (type integers or decimals)\
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\textbf{b.} tyler claims that because he started painting first, he will finish painting his side of the fence before benito finishes painting his side. is this true? explain

Explanation:

Step1: Define Variables

Let \( t \) be the time (in minutes) after Benito starts painting. Tyler's initial painted length is \( 19\frac{1}{2}=\frac{39}{2} \) feet, Tyler's rate is \( 11 \) ft/min, Benito's rate is \( 15 \) ft/min.

Step2: Set Up Equation

When Benito catches up, their painted lengths are equal:
\( 15t=\frac{39}{2}+11t \)

Step3: Solve for \( t \)

Subtract \( 11t \) from both sides:
\( 15t - 11t=\frac{39}{2} \)
\( 4t=\frac{39}{2} \)
Divide by 4:
\( t=\frac{39}{2\times4}=\frac{39}{8}=4.875 \) minutes.

Step4: Calculate Painted Length

Benito's painted length: \( 15\times4.875 = 73.125 \) feet.

Answer:

It will take \( 4.875 \) minutes. \( 73.125 \) feet will be painted on Benito’s side.

(For part b: Tyler’s total length when Benito catches up is also \( 73.125 \) feet. Tyler’s side length is \( 150 \) feet, so Tyler’s remaining: \( 150 - 73.125 = 76.875 \) feet. Benito’s remaining: \( 150 - 73.125 = 76.875 \) feet. Wait, no—wait, the fence sides are each \( 150 \) ft? Wait, the diagram shows “150 ft” for the fence. So when Benito catches up, both have painted \( 73.125 \) ft. Then, time to finish: Tyler needs \( \frac{150 - 73.125}{11}\approx6.98 \) minutes, Benito needs \( \frac{150 - 73.125}{15}=5.125 \) minutes. So Benito finishes first. So Tyler’s claim is false.)

But for part a: time is \( 4.875 \) minutes, length is \( 73.125 \) feet.