A landscaper is designing a rectangular fountain with a 4-foot-wide path around it. The equation $A = 4p + 64$ will relate the area $A$, in square feet, of the path to the perimeter $p$, in feet, of the fountain. In the design, how many feet will the perimeter of the fountain increase for each additional square foot of the path’s area?
The answer is 1/4, but I do not know how. Anyone please show me with every vital steps? Thank you!
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You have $A = 4p+64$.
If the area increases by $1$, and the new perimeter is $q$, then $A+1 = 4q+64$.
Subtracting these, $1 = (A+1)-A =(4q+64)-(4p+64) =4q-4p =4(q-p) $.
Therefore, $q-p$, which is the amount that the perimeter increased by, satisfies $q-p = \dfrac14 $.
This is where the answer comes from.
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This question, like many SAT questions, is just asking you to think about the meaning of the slope of a line.
In this case, the problem is asking: if you increase $A$ by 1, by how much does $p$ increase? Part of the test is whether you recognize that this is precisely the kind of question the slope of a line helps us answer.
You are given $A$ as a linear function of $p$, namely $A=4p+64$, so you can easily read off the slope, which is 4. But what does this mean? Saying the slope is 4 means that when $p$ increases by 1, $A$ increases by 4. In other words, the ratio
$$\frac{\text{change in A}}{\text{change in p}}$$
is 4 to 1.
Hence if we want $A$ to increase only by 1, we should increase $p$ only by $1/4$.
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There are correct and good answers here. Here's a simpler one that's also a test taking strategy. (Some regulars on this site might not appreciate this approach but I've frequently seen this strategy [successfully] taught.)
You want to know the following:
If I increase $A$ by $1$, how much will $p$ increase?
We could analyze it in general. But because there's only one right answer (which we can infer from the question's phrasing, and it's especially clear if it's a multiple choice question), we can just plug in specific and convenient values and see what happens!
What makes a value convenient? It really depends on the problem. In this case we would like to pick a value of $A$ such that solving $A = 4p+64$ for $p$ is "easy." In theory, any value of $A$ will work, but because $4p$ and $64$ are both multiplies of $4$, then a convenient value of $A$ will be any multiple of $4$ large enough to give us a positive $p$.
So let's try $A=80$. This gives us $$80 = 4p+64.$$ Solve this to get $p=4$. Now let's add $1$ to $A$ and see what happens.
If we have $A=81$, then we get $$81=4p+64.$$ So then $4p=17$, and so $$p =\frac{17}4 = 4.25 = 4+\frac14.$$
Therefore, $p$ increases by $1/4$ whenever$A$ increases by $1$.
On
The area of the path is given by $A = 4p+4\times (4\times 4)$: $4p$ makes for the four segments of path along each side of the rectangle, and you add four times a $4\times 4$ "square feet" squares to connect the segments at corners.
If you enlarge the base rectangle, you will change the perimeter, hence the surface of the four segments. But the four $4\times 4$ "square feet" squares won't change. So the surface of the path will vary only from the $4p$ term. In other words, the difference between the area for two designs varies $4$ times faster than the perimeter. This linear dependence tells you that if the perimeter varies by $1$, the area will vary by $4$, whatever the initial area. This is a proportional effect, or the slope some people here are asking for. So for the area to vary by 1, the perimeter should vary by $1/4$. We are now ready for equations.
If the surface increases by 1 square foot, i.e. $A' = A+1$, what will be the new perimeter $p'$? $A' =4p'+64$, and you know that $A =4p+64$, so $4p+64+1 = 4p'+64$, which reduces to $4(p'-p)=1$, so the perimeter variation is $p'-p =1/4$.
It is the inverse of the coefficient $4$ of $p$. If you add $1$ to $A$, you have to add $\frac 14$ to $p$ to maintain the equality.