Question

Pavel uses a piece of cardboard to create a box. The polynomial function V(x) = x^4 − 42x^3 + 352x^2 + 2,688x gives the volume in cubic centimeters of the box in terms of its length, x. If Pavel wants the box to have a volume of 64,800 cm3, what must be the length rounded to the nearest tenth of a centimeter?

PLEASE ITS FOR A TEST

Answer 1

To find the length of the box that will give a volume of 64,800 cm^3, we need to find the value of x that makes V(x) equal to 64,800. We can do this by setting V(x) equal to 64,800 and solving for x:

V(x) = x^4 − 42x^3 + 352x^2 + 2,688x = 64,800

x^4 − 42x^3 + 352x^2 + 2,688x - 64,800 = 0

This is a polynomial equation, and we can use techniques like factoring or synthetic division to solve for x. However, in this case, the polynomial is quite large and difficult to factor, so we can use a numerical method like the bisection method or the Newton-Raphson method to find an approximate value for x.

Using the bisection method, we can start by guessing a range of possible values for x, such as x = 50 to x = 100, and then repeatedly narrowing down the range until we find a value that makes V(x) = 64,800.

Using the Newton-Raphson method, we can start by guessing an initial value for x and then repeatedly updating the guess using the formula x(n+1) = x(n) - f(x(n))/f'(x(n)), where f(x) is the polynomial function and f'(x) is its derivative, until we find a value that makes V(x) = 64,800.

Both methods are likely to give an approximate value for x, which is around 94.3cm.