an electrical transformer box is a rectangular prism constructed from sheet metal. it must have a volume of at least 274 625 cm^3 to hold all the necessary equipment.
a) what dimensions for the box require the least area of sheet metal?
b) what area of sheet metal is needed to build the box?
c) Tony has 20 square feet of sheet metal. will this be enough to construct the box? justify your answer.
A cube will have the maximum volume for a minimum surface area.
length of side ---- x
x^3 = 274 625
x = 274 625^(1/3) = 65 cm
So each of the 6 sides needed = 65^2 cm^2
total surface area = 6(65)^2 cm^2 = 25,350 cm^2
20 square feet???
We were doing so nicely working in the metric system, and you come
along with that colonial unit of square feet.
Look up how many cm^2 there are in a ft^2, then find it for 20 ft^2 to see if there is enough
a) The dimensions that require the least area of sheet metal would be a cube, as a cube maximizes volume while minimizing surface area. So, all sides of the cube would have the same length. To find the length, we can take the cube root of the volume: ∛(274625) ≈ 64.13 cm.
b) The area of sheet metal needed to build the box is the sum of the surface area of all 6 sides. Since the box is a cube, each side has the same area. To find the area, we can use the formula: 6 * (side length)^2 = 6 * (64.13 cm)^2 ≈ 246179.47 cm^2.
c) To convert from cm^2 to square feet, divide the area in cm^2 by 929.0304, as there are approximately 929.0304 cm^2 in 1 square foot. So, the area in square feet is approximately 246179.47 cm^2 / 929.0304 ≈ 264.88 square feet.
Since Tony has only 20 square feet of sheet metal, it is not enough to construct the box. The amount of sheet metal needed (approximately 264.88 square feet) exceeds the available amount (20 square feet).
a) To minimize the surface area of the rectangular prism, we need to find the dimensions that minimize the sum of the areas of all six faces.
Let the length, width, and height of the rectangular prism be represented by L, W, and H, respectively.
Total volume of the rectangular prism is given as:
Volume = L * W * H = 274,625 cm^3
We need to minimize the surface area, which can be given as:
Surface Area = 2(LW + LH + WH)
To find the dimensions requiring the least area, we can start by expressing one variable in terms of the other two variables.
From the volume equation, we can express L in terms of W and H:
L = 274,625 / (W * H)
Substituting this value of L into the surface area equation, we have:
Surface Area = 2[(274,625 / WH)W + WH + H(W)] = 2[274,625 / H + WH + W^2]
Now, we can differentiate the surface area equation with respect to H and set it equal to zero to find the minimum value.
d(Surface Area) / dH = -2(274,625 / H^2) + W = 0
Simplifying, we get:
W = 2(274,625) / H^2
Similarly, we can differentiate the surface area equation with respect to W and set it equal to zero to find the minimum value.
d(Surface Area) / dW = -2(274,625 / W^2) + H = 0
Simplifying, we get:
H = 2(274,625) / W^2
Now, we have a system of equations:
W = 2(274,625) / H^2
H = 2(274,625) / W^2
Solving this system, we can find the values of W and H that minimize the surface area.
b) The area of sheet metal needed to build the box is equal to the surface area of the rectangular prism.
From the surface area equation, we have:
Surface Area = 2(LW + LH + WH)
Substituting the values of L, W, and H obtained from the previous calculation, we can find the area of sheet metal needed.
c) To determine if Tony has enough sheet metal to construct the box, we need to convert the given area of sheet metal from square feet to square centimeters (cm^2) and compare it to the calculated surface area.
If Tony has enough sheet metal, the converted area in square centimeters should be greater than or equal to the surface area of the box.
To find the dimensions for the box that require the least area of sheet metal, we need to note that a rectangular prism's volume can be calculated by multiplying the length, width, and height of the box. Since we want to minimize the surface area, which is directly related to the amount of sheet metal required, we can use the concept of the cube root of a number to find the dimensions.
a) Let's find the dimensions that minimize the surface area. The formula to calculate the volume of a rectangular prism is V = lwh, where V is the volume, l is the length, w is the width, and h is the height.
To minimize the surface area, we need to consider that the volume of the box needs to be at least 274,625 cm^3. Taking the cube root of the volume, we can find the side lengths that yield the least surface area.
Cube root of 274,625 cm^3 is approximately 64.67 cm.
Therefore, the dimensions that require the least area of sheet metal are approximately:
Length (l) = 64.67 cm
Width (w) = 64.67 cm
Height (h) = 64.67 cm
b) To calculate the surface area of sheet metal needed to build the box, we can use the formula for the surface area of a rectangular prism: SA = 2lw + 2lh + 2wh.
Substituting the given dimensions into the formula:
SA = 2(64.67 * 64.67) + 2(64.67 * 64.67) + 2(64.67 * 64.67)
SA ≈ 26,417.63 cm^2
Therefore, approximately 26,417.63 cm^2 of sheet metal is needed to build the box.
c) To determine if Tony has enough sheet metal to construct the box, we need to convert 20 square feet into square centimeters, using the fact that 1 square foot is equal to 929.03 square centimeters.
20 square feet * 929.03 square centimeters/square foot ≈ 18,580.6 square centimeters
Since Tony has 18,580.6 square centimeters of sheet metal, and the box requires approximately 26,417.63 square centimeters of sheet metal, we can conclude that Tony does not have enough sheet metal to construct the box.