Cubic Coating : Frozen specimens are stored in a cubic metal box that is x inches on each side. The box issurrounded by a 2 inch thick layer of foam insulation.(i) Find a polynomial function ( )V x that gives the total volume in cubic inches for the box and insulation.(ii) Find the total volume if x is 10 inches.(iii) Use the remainder theorem to find the total volume when x is 10 inches.
Question
Cubic Coating : Frozen specimens are stored in a cubic metal box that is x inches on each side. The box issurrounded by a 2 inch thick layer of foam insulation.(i) Find a polynomial function ( )V x that gives the total volume in cubic inches for the box and insulation.(ii) Find the total volume if x is 10 inches.(iii) Use the remainder theorem to find the total volume when x is 10 inches.
Solution
(i) To find the polynomial function V(x) that gives the total volume in cubic inches for the box and insulation, we need to calculate the volume of the box and the volume of the foam insulation separately.
The volume of the box is given by V_box = x^3, since each side of the box is x inches.
The volume of the foam insulation can be calculated by subtracting the volume of the inner box from the volume of the outer box. The outer box has dimensions (x + 4) inches on each side (2 inches of foam insulation on each side), so its volume is V_outer = (x + 4)^3. The inner box has dimensions x inches on each side, so its volume is V_inner = x^3.
Therefore, the volume of the foam insulation is V_foam = V_outer - V_inner = (x + 4)^3 - x^3.
The total volume V(x) is the sum of the volume of the box and the volume of the foam insulation, so we have:
V(x) = V_box + V_foam = x^3 + (x + 4)^3 - x^3.
Simplifying this expression, we get:
V(x) = (x + 4)^3.
(ii) To find the total volume when x is 10 inches, we substitute x = 10 into the polynomial function V(x):
V(10) = (10 + 4)^3 = 14^3 = 2744 cubic inches.
Therefore, the total volume when x is 10 inches is 2744 cubic inches.
(iii) To use the remainder theorem to find the total volume when x is 10 inches, we divide the polynomial function V(x) by (x - 10) and find the remainder.
Using long division or synthetic division, we divide (x + 4)^3 by (x - 10):
14
_______
x - 10 | (x + 4)^3
The remainder is 2744, which is the same as the total volume when x is 10 inches.
Therefore, using the remainder theorem, the total volume when x is 10 inches is 2744 cubic inches.
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