Video Transcript
For a rectangle of fixed area, the length 𝑙 varies inversely with its width 𝑤. Given that 𝑙 equals 22 centimeters when 𝑤 equals 16 centimeters, determine the value of 𝑙 when 𝑤 equals 44.
We’re told that the length 𝑙 varies inversely with the width 𝑤. In other words, 𝑙 is inversely proportional to 𝑤. If 𝑙 is inversely proportional to 𝑤, we say that 𝑙 is directly proportional to one over 𝑤. And the corresponding equation we can use here is 𝑙 equals 𝑘 divided by 𝑤 for some real constant 𝑘. We think about this as, as 𝑤 increases, the value of 𝑙 decreases, and vice versa.
And this makes a lot of sense because we want a rectangle to have a fixed area. If we increase the length, we have to decrease the width, and vice versa, to ensure that the area remains unchanged. So we can think about this constant 𝑘 as actually being the fixed area 𝐴 of the rectangle.
So now we have an equation. Let’s use the fact that when 𝑙 equals 22, 𝑤 is 16. And this will allow us to find the value of 𝑘. Our equation becomes 22 equals 𝑘 over 16. And we solve for 𝑘 by multiplying through by 16. So 𝑘 is 22 times 16. And remember, if we think about this geometrically, we said 𝑘 is the area of our rectangle. So it makes a lot of sense that we would be multiplying the length by the width. And so 𝑘 is equal to 352 or 352 square centimeters.
So, now thinking about our earlier equation, we’re now able to substitute 𝑘 equals 352 to form an equation that links 𝑙 and 𝑤 for all fixed areas of 352. It’s 𝑙 equals 352 over 𝑤. And this is great because we want to find the value of 𝑙 when 𝑤 is 44. We substitute 𝑤 is 44 into the equation, and we get 𝑙 equals 352 divided by 44, which of course is equal to eight. So the value of 𝑙 when 𝑤 is equal to 44 is eight centimeters.
Now, where possible, we should try and check our answer. We thought about this geometrically as well as algebraically. We said that 𝑘 was the value of the area, and area of a rectangle is length times width. So let’s think about the rectangle we generated. It has a length of eight centimeters and a width of 44. We should check that eight times 44 gives us that area of 352. Well, eight times 44 is 352. So this is a good indication that we’ve likely done our calculations correctly.
