The maximum volume of the rectangular prism is 7500 cubic units when x=5.
The volume V of a rectangular prism is given by the formula:
V=Base Area×Height
The base area is given by the polynomial 36+15x−3x^2 , and the height is 20x−90. To maximize the volume, we'll first express the volume function and then find its critical points.
V(x)=Base Area×Height
V(x)=(36+15x−3x^2 )×(20x−90)
V(x)=(36+15x−3x^2 )×(20x−90)
V(x)=(3x^2 −15x−36)×(20x−90)
V(x)=−60x^3 +360x^2 −900x+1800x−5400
V(x)=−60x^3 +360x^2 +900x−5400
Now, to find the maximum volume, take the derivative of V(x) and set it to zero to find critical points.
V′(x)=−180x^2 +720x+900
Setting V′ (x) to zero:
−180x^2 +720x+900=0
Dividing the equation by -180 to simplify:
x^2 −4x−5=0
Factoring:
(x−5)(x+1)=0
Therefore, the critical points are x=5 and x=−1.
To find which value of x gives the maximum volume, you'll need to test these critical points and the endpoints of the interval where x lies. Since x represents a length, and lengths can't be negative, we consider x=5.
At x=5, the volume function V(x)=−60x^3 +360x^2 +900x−5400 will give the maximum volume of the rectangular prism. Plug x=5 into V(x) to find the maximum volume.
V(5)=−60(5)^3 +360(5)^2 +900(5)−5400
V(5)=−60(125)+360(25)+4500−5400
V(5)=−7500+9000+4500−5400
V(5)=7500
Hence, the maximum volume of the rectangular prism is 7500 cubic units when x=5.