One side of a box has a length of 𝑙 is equal to 150 centimeters and an area of 6500 centimeters squared, as shown in the diagram. What is the height ℎ of the box to the nearest centimeter?
Okay, so in this question, we’re looking at one particular side, one particular face, of a box where the box itself may look something like this. But we are only interested in the particular face that we’re currently looking at, this one here. And this particular face of the box has a length 𝑙 and a height ℎ. We know that the length 𝑙 is equal to 150 centimeters. We’ve been told this in the question. And we’ve been asked to find the height ℎ. And to do this, we’ve been given the information that the area of the face of the box — we’ll call this area capital 𝐴 — is equal to 6500 centimeters squared. And this is important because we can realize that the face of the box has a rectangular shape. It’s a rectangle.
And therefore, we can recall that the area of a rectangle, which once again we will call 𝐴, is given by multiplying the length of the rectangle by the height of the rectangle. Therefore, if we want to find the height of the rectangle, we simply need to rearrange this equation. So we start by saying that the area is equal to the length multiplied by the height. And then to rearrange, we want to try and isolate the height on one side of the equation. We do this by dividing both sides of the equation by the length 𝑙. Because this way, on the right-hand side, we have the height multiplied by the length divided by the length. And when you multiply and divide by the same thing, that’s simply the same as multiplying by one.
And so, on the right-hand side, we’re simply left with the height ℎ. Whereas on the left, we’ve got the area divided by the length 𝑙. And since we know both of the quantities on the left-hand side of the equation, we can simply substitute them in. And when we do so, the fraction simply becomes 6500 centimeters squared, which is the area, divided by 150 centimeters, which is the length 𝑙. Now, before we evaluate this fraction, let’s quickly think about what happens to the units of this quantity. Because in this case the fraction consists of an area divided by a length. And so, it should have the units of another length because what we’re measuring here is the height of the box ℎ. But the height is also another length.
And that’s exactly what is going to happen. Because in the numerator, we’ve got centimeter squared or square centimeters. And square centimeters are the same thing as a unit of centimeters multiplied by another unit of centimeters. Whereas in the denominator, we simply got one power of centimeters, which means that one power of centimeters in the numerator cancels with the one power of centimeters in the denominator, leaving us with an overall power of centimeters in the numerator.
In other words, the overall fraction here is going to have the unit centimeters, which is the unit of a length. And that’s exactly what we need. So now, all that’s left to do is to evaluate the number 6500 divided by 150. And so, when we do evaluate the entire fraction, what we find is that the height ℎ is equal to 43.3333 recurring centimeters. However, remember, we’ve been asked to give our answer to the nearest centimeter. And in order to do this, we’re going to have to round our value at this position here.
But in order to work out what happens to this last number in our rounded answer, we need to look at the next number. Now, this number happens to be a three and three is less than five. Therefore, the last number in our rounded value will stay exactly the same. It’s not going to round up. In other words, to the nearest centimeter then, our value is 43 centimeters.
And hence, we can say that the side of the box with a length 𝑙 of 150 centimeters and an area of 6500 centimeters squared has a height ℎ of 43 centimeters to the nearest centimeter.