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Problem 15 Hard Difficulty

A homeowners’ association limits the dimensions of the pools that it will allow in a particular subdivision. The bylaws state that permits will only be granted for pools shaped like rectangular prisms, for which the sum of the length of the pool and the perimeter of the vertical side containing the ladder cannot exceed 200 meters. The perimeter of the ladder side is determined using the width and the depth of the pool. If a pool has a length of 75 meters and its width is 1.5 times its depth, which of the following shows the allowable depth a, in meters, of the pool?
A) $0 < a \leq 62 \frac{1}{2}$
B) $0 < a \leq 50$
C) $0 < a \leq 31 \frac{1}{4}$
D) $0 < a \leq 25$

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Video Transcript

Alright. Sometimes an information is given like this, it's just easier to draw it out so I'm just gonna drop my pool. Okay? And my length in this case the problem tells me is 75 and it says my width, which is um This part of the pool, we're not sure what it is. Other than that it's 1.5 times the depth. So w equals 1.5 a. And then this is the depth of the pool, right? I don't know how deep it is. And the problem tells us that the perimeter Plus the length of the pool cannot exceed 200. Okay? So whenever we find the perimeter of this side of our rectangle, we would do two W. Pleiss P. A. Plus the length, Sorry to w. plus two a. Plus the length. And it can't be greater than 200. All right, well, we know that our W. Is 1.5 A. So I could just use substitution and plug in 1.5 A. Into that W. And we would get two times 1.5 A. Plus to a plus. Uh And then we can just continue solving for the depth. So I could do um multiply this two out to get rid of the parentheses in two times 1.5 a. Is three a. And we could then just combine like terms we have that. Sorry, I forgot that pull a sign there. We have that three A. Plus at two A. Which is five A. Have that length still. Mhm. All right. And then this problem told us the length was 75. So again I could just plug in that 75 right there and I've got five a. Plus 75 cannot be greater than or equal to 200. And then I could continue solving for a. So I can move that 75 over to the other side. and whenever we do that we have five a cannot be greater than or equal to 1 25. And then we could divide both sides by five to get a by itself and we get a It's less than or equal to 25. All right. Now to line up with the answer choices, there's a zero in front of it. So you have zero is less than or equal a. Is greater than or equal to zero. And that's because it's a real real world problem. Right? Our pool cannot be a negative number. So this is what your final answer will look like. So your answer is going to be d. Yeah.