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)
Reiny, you were correct until the last segment of your expression for the V-SA Ratio. It is in fact...
x(100-2x)(100-2x)/(100-2x)(100+2x) And then the (100-2x) cancel formulating
V/SA= x(100-2x) / (100+2x)
Thanks,Reiny for all your help with my math!
welcome
a) To express the volume of the box as a function of x, we need to find the product of the length, width, and height of the box. The length and width of the base are given by the expression (100 - 2x), and the height is x. Therefore, the volume V can be expressed as:
V = (100 - 2x)(100 - 2x)(x)
V = (100 - 2x)^2(x)
b) To express the surface area of the box as a function of x, we need to find the sum of the areas of each side of the box. The area of each side can be calculated by multiplying the length and width. There are six sides in total, but since the box has no lid, we subtract the area of the top (which is equal to the area of the base). Therefore, the surface area S can be expressed as:
S = 2[(100 - 2x)(100 - 2x)] + 4(100 - 2x)(x)
S = 2(100 - 2x)^2 + 4(100 - 2x)(x)
c) The ratio of the volume of the box to its surface area can be expressed as:
R = V / S
R = [(100 - 2x)^2(x)] / [2(100 - 2x)^2 + 4(100 - 2x)(x)]
To simplify this expression, we can factor out common terms from the numerator and denominator:
R = [(100 - 2x)^2(x)] / [2(100 - 2x)^2 + 4(100 - 2x)(x)]
R = [(100 - 2x)^2(x)] / [2(100 - 2x)(100 - 2x + 2x)]
Now we can cancel out common factors in the numerator and denominator:
R = [(100 - 2x)(100 - 2x)(x)] / [2(100 - 2x)(100)]
R = (100 - 2x)(x) / 200
The non-permissible values of x would be any values that make the denominator zero, as division by zero is undefined. In this case, when 100 - 2x = 0, we have:
2x = 100
x = 50
So x = 50 is a non-permissible value in the expression for the ratio of volume to surface area.
The have given you everything you have to know in the description.
so since V = lxwxh
= (100-2x)(100-2x)(x)
= x(100-2x)^2
SA = base + 4 sides
= (100-2x)^2 + 4x(100-2x)
= (100-2x)[ 100-2x + 4x]
= (100-2x)(100+2x)
ratio of V/SA
= x(100-2x)/( (100-2x)(100+2x) )
= x/(100+2x) , x ≠ 50