An open-top box is made from a 14-inch-by-32-inch piece of cardboard, as shown below. The volume of the box is represented by V(x) = x(14 - 2x)(32 - 2x), where x is the height of the box.
a. Write the volume of the box as a polynomial function in standard form.
b. Find the volume of the box if the height is 2 inches.
a. Okay, let's solve this puzzle. We have the volume function V(x) = x(14 - 2x)(32 - 2x). To get it into standard form, we need to multiply everything out and combine like terms.
So, V(x) = x(14 - 2x)(32 - 2x)
= x(448 - 28x - 64x + 4x^2)
= 4x^3 - 92x^2 + 448x
Boom! We have the volume of the box as a polynomial function in standard form.
b. Now that we have our function, let's plug in x = 2 to find the volume of the box when the height is 2 inches.
V(2) = 4(2)^3 - 92(2)^2 + 448(2)
= 4(8) - 92(4) + 448(2)
= 32 - 368 + 896
= 560
Voila! The volume of the box when the height is 2 inches is 560 cubic inches.
a. To write the volume of the box as a polynomial function in standard form, we need to expand and simplify the expression V(x) = x(14 - 2x)(32 - 2x).
First, let's expand the expression using the distributive property:
V(x) = x(14)(32) - x(14)(2x) - x(2x)(32) + x(2x)(2x)
= 448x - 28x^2 - 64x^2 + 4x^3
Now, let's combine like terms and rewrite it in standard form:
V(x) = 4x^3 - 92x^2 + 448x
b. To find the volume of the box if the height is 2 inches, we substitute x = 2 into the polynomial function V(x):
V(2) = 4(2)^3 - 92(2)^2 + 448(2)
= 4(8) - 92(4) + 896
= 32 - 368 + 896
= 560 cubic inches
a. To write the volume of the box as a polynomial function in standard form, we can start by expanding the expression V(x) = x(14 - 2x)(32 - 2x).
First, distribute x to both terms inside the parentheses:
V(x) = x * (14 * (32 - 2x) - 2x * (32 - 2x))
Simplify each term:
V(x) = x * (448 - 28x - 64x + 4x^2)
Combine like terms:
V(x) = x * (4x^2 - 92x + 448)
Finally, multiply x by each term in the parentheses:
V(x) = 4x^3 - 92x^2 + 448x
So, the volume of the box as a polynomial function in standard form is V(x) = 4x^3 - 92x^2 + 448x.
b. To find the volume of the box when the height is 2 inches, we substitute x = 2 in the polynomial function V(x) = 4x^3 - 92x^2 + 448x.
V(2) = 4(2)^3 - 92(2)^2 + 448(2)
V(2) = 4(8) - 92(4) + 448(2)
V(2) = 32 - 368 + 896
V(2) = 560 cubic inches
So, the volume of the box when the height is 2 inches is 560 cubic inches.
(a) just expand the polynomial.
(b) v = 2(14-4)(32-4) = 672 in^3