A particular company needs to ship cubed shaped boxes of varying sizes to customers. As these boxes are fragile they are to be packed inside a larger box with a 5 cm clearance on each side to allow space for a packing material. If v(x) is the volume of the smaller box as a function of its edge length, x, in centimetres, determine an expression for the volume of the larger box as a transformation of the function v. Can someone show me the full step by step workouts. Thanks
To determine the expression for the volume of the larger box, we need to consider the dimensions of the smaller box along with the clearance on each side.
Let's start by finding the volume of the smaller box as a function of its edge length, x. We can express this as v(x) = x^3, where x represents the length of one side of the smaller box.
Now, we need to calculate the edge length of the larger box. Since there is a 5 cm clearance on each side, the length of each side of the larger box will be x + 2(5 cm) = x + 10 cm.
To find the volume of the larger box, we need to cube this length, so the volume of the larger box, V(x), can be expressed as:
V(x) = (x + 10 cm)^3
Expanding this expression, we have:
V(x) = (x + 10 cm)(x + 10 cm)(x + 10 cm)
= (x + 10 cm)(x^2 + 20x + 100 cm^2)
= x(x^2 + 20x + 100 cm^2) + 10 cm(x^2 + 20x + 100 cm^2)
= x^3 + 20x^2 + 100x cm^2 + 10x^2 + 200x + 1000 cm^3
= x^3 + 30x^2 + 300x + 1000 cm^3
Therefore, the expression for the volume of the larger box, V(x), as a transformation of the function v, is V(x) = x^3 + 30x^2 + 300x + 1000 cm^3.
This expression represents the volume of the larger box when the edge length of the smaller box is x centimeters.