Suppose the height of the freight elevator in your building is half its width w

when the doors are all the way open.
a. What is an expression for the maximum side length of a sheet of metal that will fit through the elevator doors?
b. If the height of the elevator is 3 meters, what is the maximum length that
will fit through the doors?

the doorway is h by w

a. a thin sheet of metal could be as long as the diagonal, √(h^2+w^2)
b. Now just plug in your numbers

a. Let's denote the height of the freight elevator as h and the width as w. We are given that the height is half the width when the doors are all the way open, so we can write the expression for the maximum side length of a sheet of metal that will fit through the elevator doors as:

Maximum side length = h/2

b. If the height of the elevator is 3 meters, we can substitute this value into the expression for the maximum side length:

Maximum side length = 3/2 = 1.5 meters

Therefore, the maximum length that will fit through the doors is 1.5 meters.

To find the expression for the maximum side length of a sheet of metal that will fit through the elevator doors, we can start by considering the given information that the height of the elevator is half its width when the doors are fully open.

Let's denote the width of the elevator as w. According to the given information, the height of the elevator is half of w, so the height would be w/2.

a. Expression for the maximum side length of a sheet of metal:
To determine the maximum side length of the sheet of metal that can fit through the elevator doors, we need to consider that the sheet should not be longer than the height or width of the elevator opening. Therefore, the maximum side length would be the smaller value between the height and the width of the elevator opening.

Since the height of the elevator is w/2, the expression for the maximum side length of the sheet of metal would be min(w, w/2).

b. If the height of the elevator is 3 meters:
If we know that the height of the elevator (w/2) is 3 meters, we can substitute this value into the expression to find the maximum length that will fit through the doors.

max side length = min(w, w/2)
Substituting the height value:
max side length = min(w, 3)

Note that without knowing the specific value of the width (w), we cannot determine the exact maximum length. We can only conclude that the maximum side length will be the smaller value between the width and half of the width (3 in this case).