If two opposite sides of a rectangle increase in length, how m...

Question

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

Accepted Answer

Welcome back, everyone. In a rectangular swimming pool, if the land is raised, how should the weds be modified to keep the pool's area constant? We given for answer choices A says the width should be increased, B decreased, C unchanged, and D, more information is needed. So let's understand that we are looking at a rectangular swimming pool, which is going to have an area of A equals with W multiplied by length L, right? And we want to make sure that it is constant. So let's consider two conditions. Condition number 1, we have a specific width W1 multiplied by a specific length L1. And now we're going to set it equal to our second condition. Let's suppose that we are changing the length to some value L2, and this means that there is going to be a new width, but we're setting them equal to each other because area 1 is going to be equal to area 2, right? We want to make sure that the area is constant. And now we are looking at the new width. So what we want to do is simply solve for. W2, right? We want to solve for the new width and see that we are dividing W1 L1 by L2. What we are showing here is that this is a function, right? So the new width. Relative to, we can essentially say that with 2 is equal to. A constant, let's remember that our numerator is a constant. It, it is area, right? So let's suppose that it is K. Which is a constant. And our L2 varies relative to W2. So we can say that W2 is equal to some constant K divided by L2, and this is basically functions. We can say W2 is a function of L2 because our numerator involves a constant k. So we get W2 of L2 is K divided by L2, and we can recall that this has a form of Y equals K divided by X, right? Which essentially indicates an inverse proportionality. So if we're looking at the width, we have to understand that if L2 increases, understanding that the numerator is a constant, then W2 decreases. And vice versa if L1 decreases based on the relationship of an inverse proportionality, we can say that W2 is going to increase. So now in this context we are asked how should the width be modified to keep the pool area constant if the length is increased. So, because this is an inverse relationship, we understand that the width should be decreased, which corresponds to the answer choice B. Thank you for watching.

If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

Key Concepts

Area of a Rectangle

Inverse Relationship

Proportionality

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