Kirkor has to design and build a box with the greatest volume possible. the box is a rectangular prism. for each surface area, what will be the dimensions of the box?
a) 600 square inches
b) 1350 cm^2
c) 2400 square inches
a)
600=6A^2
600/6=(6A^2)/6
100=A^2
√100=√(A^2)
10=A
b)
1350=6A^2
1350/6=(6A^2)/6
1225=A^2
√225=√(A^2)
15=A
c)
2400=6A^2
2400/6=(6A^2)/6
400=A^2
√400=√(A^2)
20=A
is my answer for a,b and c correct?
The greatest volume is obtained if the box is a cube
let each side length be x
Each side will have a surface area of x^2 and there are 6 such sides
a) 6x^2 = 600
x^2 = 100
x = 10
the box will be 10 by 10 by 10
Do the other two in the same way
b)
1350=6A^2
1350/6=(6A^2)/6
1225=A^2
√225=√(A^2)
15=A
c)
2400=6A^2
2400/6=(6A^2)/6
400=A^2
√400=√(A^2)
20=A
To find the dimensions of the box with the greatest volume possible for each given surface area, we need to consider the formula for the volume of a rectangular prism. The volume (V) of a rectangular prism is given by V = length (L) × width (W) × height (H).
a) Surface area = 600 square inches:
First, let's find the dimensions that will give us a surface area of 600 square inches. The surface area (SA) of a rectangular prism is given by the formula SA = 2(LW + LH + WH).
We can set up the equation:
2(LW + LH + WH) = 600
Simplifying the equation a bit, we get:
LW + LH + WH = 300
Since we want to maximize the volume, we can assume the box is a cube, where L = W = H. Substituting L = W = H into the equation, we get:
3L^2 = 300
Simplifying further, we find:
L^2 = 100
L = 10 inches
Therefore, the box with a surface area of 600 square inches will have dimensions of 10 inches × 10 inches × 10 inches.
b) Surface area = 1350 cm^2:
Similar to the previous case, we use the formula for the surface area to set up the equation:
2(LW + LH + WH) = 1350
Assuming the box is a cube, we have L = W = H, giving us:
3L^2 = 1350
Simplifying, we find:
L^2 = 450
L ≈ 21.2 cm
Therefore, the box with a surface area of 1350 cm^2 will have dimensions of approximately 21.2 cm × 21.2 cm × 21.2 cm.
c) Surface area = 2400 square inches:
Again, we set up the equation using the formula for surface area:
2(LW + LH + WH) = 2400
Assuming the box is a cube, we have L = W = H, resulting in:
3L^2 = 2400
Simplifying this equation, we find:
L^2 = 800
L ≈ 28.3 inches
Thus, the box with a surface area of 2400 square inches will have dimensions of approximately 28.3 inches × 28.3 inches × 28.3 inches.
Note: The dimensions given in all the cases assume that the box is a rectangular prism with equal sides, maximizing the volume.