a closed box has a base twice as long as it is wide.if its volume is 100 in cubed,express its surface area as a function of the width x of its base.
To express the surface area of the closed box as a function of the width x of its base, we need to find the dimensions of the box first.
Let's assume that the width of the base is x, then the length of the base will be 2x since the base is twice as long as it is wide.
The formula for the volume of a rectangular box is V = lwh, where V is the volume, l is the length, w is the width, and h is the height. In this case, we have the volume V = 100 in^3. We also know that the length of the base is 2x.
So, we can set up the equation:
100 = (2x)(x)(h)
Now, let's solve for h in terms of x. Divide both sides of the equation by 2x:
50/x = h
Therefore, the height of the box is equal to 50/x.
Now, we can calculate the surface area of the box. The formula for the surface area of a rectangular box is A = 2lw + 2lh + 2wh, where A is the surface area, l is the length, w is the width, and h is the height.
Substituting the known values into the formula, we get:
A = 2(2x)(x) + 2(2x)(50/x) + 2(x)(50/x)
Simplifying this expression, we get:
A = 4x^2 + 200/x + 100
Therefore, the surface area A of the closed box can be expressed as a function of the width x of its base as A(x) = 4x^2 + 200/x + 100.
To find the surface area of the closed box, we need to identify all sides and calculate their areas.
Let's assume the width of the base of the box is x. Since the length of the base is twice the width, the length would be 2x.
The height of the box cannot be determined from the given information, so we can represent it as h.
1. Top and Bottom Surfaces:
The top and bottom surfaces of the box would have dimensions of length x and width 2x. Therefore, their areas are:
2(x)(2x) = 4x^2 each.
2. Front and Back Surfaces:
The front and back surfaces have dimensions of length 2x and height h. Thus, their areas are:
2(2x)(h) = 4xh each.
3. Side Surfaces:
The side surfaces have dimensions of length x and height h. Hence, their areas are:
2(x)(h) = 2xh each.
To calculate the total surface area, we add up all the individual areas:
Total Surface Area = 2(4x^2) + 2(4xh) + 2(2xh)
= 8x^2 + 8xh + 4xh
= 8x^2 + 12xh
Therefore, the surface area of the closed box is given by the function:
Surface Area = 8x^2 + 12xh