A square sheet of cardboard 100 cm by 100 cm is to have corners of side length x cut out. These cutouts will create flaps that can be folded up to form a box with no lid. The length and width of the base of the box are given by the expression (100 – 2x), and the height of the box is x, where x is in centimetres.
a) Express the volume of the box as a function of x. (2 marks)
b) Express the surface area of the box as a function of x. (2 marks)
c) What is a simplified expression for the ratio of the volume of the box to its surface area? Identify any non-permissible values of x. (2 marks)
My work for a,b,c:
a)(100-2x)x would be function of volume with a x.
b)I'm not sure about this one, I need help here
c) Same here because I'm not sure and I need help
If anyone could please help me and show me where I went wrong with this, I will be extremely grateful for your help with my math!
(a) You need 3 dimensions for volume. What you have is the area of one of the sides of the box. You need
(100-2x)(100-2x)x = x(100-2x)^2
(b) You need the area of the base, plus the area of the 4 sides:
(100-2x)^2 + 4x(100-2x) = (100-2x)(100+2x)
(c) Now that you have the two expressions, the ratio is
x(100-2x)^2 / (100-2x)(100+2x) = x/(100+2x)
a) Expressing the volume of the box as a function of x:
To find the volume of the box, we need to multiply the length, width, and height together.
Length = 100 - 2x (since the cutouts decrease each side by x cm)
Width = 100 - 2x (same reason as length)
Height = x (given)
Volume = Length x Width x Height
Volume = (100 - 2x)(100 - 2x)(x)
Volume = (100 - 2x)^2 * x
So the volume of the box can be expressed as a function of x as V(x) = (100 - 2x)^2 * x.
b) Expressing the surface area of the box as a function of x:
The surface area of the box consists of the area of its six faces.
The top and bottom faces have the dimensions (100 - 2x) x (100 - 2x).
So the area of each top/bottom face is (100 - 2x)(100 - 2x).
The four side faces have dimensions x x (100 - 2x).
So the area of each side face is x(100 - 2x).
Total Surface Area = 2 * (area of top face) + 4 * (area of side face)
Surface Area = 2 * (100 - 2x)(100 - 2x) + 4 * x(100 - 2x)
Surface Area = 2(10000 - 400x + 4x^2) + 4(100x - 2x^2)
Surface Area = 20000 - 800x + 8x^2 + 400x - 8x^2
Surface Area = 20000 - 400x
So the surface area of the box can be expressed as a function of x as A(x) = 20000 - 400x.
c) Simplified expression for the ratio of volume to surface area:
To find the ratio of the volume of the box to its surface area, we divide the volume by the surface area.
Ratio = V(x) / A(x)
Ratio = [(100 - 2x)^2 * x] / [20000 - 400x]
To simplify this expression, we can cancel out common factors in the numerator and denominator.
Ratio = [x(100 - 2x)(100 - 2x)] / [20000 - 400x]
Ratio = [x(100 - 2x)(100 - 2x)] / [400(50 - x)]
Non-permissible values of x would be where a denominator equals zero.
In this case, when 400(50 - x) = 0, which means (50 - x) = 0, so x = 50.
Therefore, the non-permissible value for x is x = 50.
So, the simplified expression for the ratio of volume to surface area, with the non-permissible value of x excluded, is: [x(100 - 2x)(100 - 2x)] / [400(50 - x)], for x ≠ 50.