Question

4. VOLUME The volume of one column of the Lincoln Memorial is approximated by the expression (x³ +37x²-110x + 400). If the height of the column is x + 40 feet, find the expression that approximates the area of the base of the column in terms of x and pi.

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Answer 1

To find the area of the base of the column, we need to find the expression for the radius of the column, r, in terms of x.

The volume of a column can be calculated as V = πr²h, where r is the radius, h is the height, and π is pi.

Using the given expression for the volume of the column, we can write:

πr²h = x³ + 37x² - 110x + 400

Since h = x + 40, we can substitute to get:

πr²(x + 40) = x³ + 37x² - 110x + 400

Dividing both sides by π(x + 40) gives us:

r² = (x³ + 37x² - 110x + 400) / (π(x + 40))

Finally, taking the square root of both sides gives us the expression for the radius in terms of x:

r = √((x³ + 37x² - 110x + 400) / (π(x + 40)))

So, the expression that approximates the area of the base of the column in terms of x and π is:

A = πr² = π √((x³ + 37x² - 110x + 400) / (π(x + 40)))²