A rectangular box has a perimeter of 50 inches. Find the length and width of the box that would give the maximum area. Find the maximum possible area of the box.
I'm random.
I think the answer is 12.5 inches but this problem is probably way harder than I thought lol
square :)
side = 50/4 = 25/2
A = x y
P = 2x+2y = 50
so y = (25-x)
A = x(25-x) = 25 x - x^2
x^2 -25 x = -A
x^2 - 25 x + (25/2)^2 = -A + 625/4
x = 25/2 sure enough so y = 25/2
and A = 625/4 = (25/2)^2
To find the dimensions that would give the maximum area, we can use the fact that the perimeter of a rectangle is given by the equation:
Perimeter = 2 * (length + width)
In this case, the perimeter is given as 50 inches. So we can write the equation as:
50 = 2 * (length + width)
Divide both sides of the equation by 2:
25 = length + width
Now we need to express the area of the rectangle in terms of a single variable, either length or width. The area of a rectangle is given by the equation:
Area = length * width
We can express length in terms of width using the equation above and the one we derived earlier:
length = 25 - width
Now substitute the expression for length into the equation for area:
Area = (25 - width) * width
Simplify the equation:
Area = 25w - w^2
To find the maximum possible area, we can take the derivative of the equation with respect to width and set it equal to zero:
d(Area)/dw = 25 - 2w = 0
Solve for w:
25 - 2w = 0
2w = 25
w = 25/2 = 12.5
Now substitute w = 12.5 into the equation for length:
length = 25 - width = 25 - 12.5 = 12.5
So, the width of the box that gives the maximum area is 12.5 inches, and the length is also 12.5 inches.
To find the maximum possible area, substitute the values of width and length into the equation for area:
Area = length * width = 12.5 * 12.5 = 156.25 square inches
Therefore, the maximum possible area of the box is 156.25 square inches.
To find the length and width of the box that would give the maximum area, we need to use the fact that the perimeter of a rectangle is given by the formula P = 2l + 2w, where "l" represents the length and "w" represents the width. In this case, we are given that the perimeter is 50 inches, so we can write this as:
50 = 2l + 2w
To find the maximum possible area, we need to maximize the product of the length and width. The formula for the area of a rectangle is given by the formula A = lw.
To find the maximum area, we can use the fact that if we express one variable in terms of the other, we can substitute it into the area formula. From the perimeter equation, we can solve for l in terms of w:
50 = 2l + 2w
Subtract 2w from both sides:
50 - 2w = 2l
Divide both sides by 2:
25 - w = l
Now, substitute this expression for l into the area formula:
A = lw
A = (25 - w)w
To find the maximum area, we can take the derivative of the area formula and set it equal to 0, then solve for w:
dA/dw = 25 - 2w = 0
Solving for w, we find that w = 12.5.
Substituting this value back into the expression for l, we find that l = 12.5 as well.
Therefore, the length and width that would give the maximum area for the rectangular box are both 12.5 inches.
To find the maximum possible area, we can substitute these values into the area formula:
A = lw = 12.5 * 12.5 = 156.25 square inches.
So, the maximum possible area of the box is 156.25 square inches.