a rectangular box with a volume of 60 ft^3 fas a square base. Find a funtion that models its surface area S in terms of the length x of one side of its base
Since the base is square, the base area is x^2. The Volume is
x^2 * h = 60, where h is the height.
The surface area is 2x^2 + 4xh
= 2x^2 + 4x*(60/x^2) = 2x^2 + 240/x
Well, first we need to find the height of the box based on its volume. Let's call the length of one side of the base "x" and the height "h". Since the box has a square base, the base area is x^2, and the volume of the box is x^2 * h.
Given that the box has a volume of 60 ft^3, we can solve for the height:
x^2 * h = 60
Now, we can express the surface area of the box. It consists of the area of the four sides plus the area of the top and the bottom. The area of the top and the bottom will be 2 times the area of the base.
S = 2(x^2) + 4(x * h)
Since we already determined that x^2 * h = 60, we can substitute it into the equation:
S = 2(x^2) + 4(x * 60/x^2)
S = 2x^2 + 4(60x/x^2)
S = 2x^2 + 240/x
Now you have a function S that models the surface area of the rectangular box in terms of the length of one side of its base. But hey, don't be too impressed by all these numbers—it's just a fancy way to say, "The bigger the box, the more wrapping paper you need!"
No clowning around when it comes to math.🤡
To find a function that models the surface area (S) of a rectangular box with a square base in terms of the length (x) of one side of its base, we need to break down the surface area into its component parts.
First, let's consider the six faces of the rectangular box. The box has two bases, each with an area of x^2 (since it is a square base). So, the total area of the bases is 2x^2.
Next, the box has four sides, each with an area of x times the height (h) of the box. Since the volume of the box is given as 60 ft^3, we can write the equation for volume as V = x^2 * h.
If we solve this equation for h, we get h = 60/x^2.
Now, we can calculate the area of all four sides: 4 * (x * h) = 4 * (x * 60/x^2) = 240/x.
Finally, we can express the total surface area (S) of the rectangular box in terms of x by adding the areas of the bases and the four sides:
S = 2x^2 + 240/x.
Therefore, the function that models the surface area S in terms of the length x of one side of its base is:
S(x) = 2x^2 + 240/x.