a rectangular box has a volume of V(x)=x^3+13x^2+50x+56 cubic inches. The height of the box is x+7 inches. The width of the box is x+4 inches. Find the length of the box in terms of x.
Divide the cubic equation by (x+7)(x+4), using polynomial long division, or by factoring the cubic.
Hint: In this case, (x+4) and (x+7) are factors of the cubic. Find the third factor.
Second hint:
(x^3 +13x^2 +50x + 56)/(x+4)
= x^2 + 9x + 14
(Now, factor that)
x+2 inches
To find the length of the box in terms of x, we need to consider the formula for the volume of a rectangular box:
Volume (V) = Length x Width x Height.
Given that the volume is given by V(x) = x^3 + 13x^2 + 50x + 56, the height is x + 7, and the width is x + 4, we can substitute these values into the volume formula:
V(x) = (Length) x (Width) x (Height).
(x^3 + 13x^2 + 50x + 56) = (Length) x (x + 4) x (x + 7).
By simplifying and expanding the equation, we get:
x^3 + 13x^2 + 50x + 56 = (x^2 + 11x + 28) x (x + 7).
Expanding the right side further:
x^3 + 13x^2 + 50x + 56 = (x^3 + 7x^2 + 28x) + (11x^2 + 77x + 196).
Combining like terms:
x^3 + 13x^2 + 50x + 56 = x^3 + 7x^2 + 28x + 11x^2 + 77x + 196.
Canceling out the common terms on both sides:
0 = 4x^2 + 99x + 140.
Now we need to solve this equation to find the value of x. Once we have x, we can substitute it back into the equation x + 4 to find the width, and x + 7 to find the height.
To solve a quadratic equation like 4x^2 + 99x + 140 = 0, we can use factoring, completing the square, or the quadratic formula.