Question

Archimedes (287-212 B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times, discovered that the area under a parabolic arch like the one shown here is always two- thirds the base times the height. (a) Find the area under the parabolic arch \( y=6-x-x^2, -3 \leq x \leq 2 \) (b) Find the height of the arch. (c) Show that the area is two-thirds the base times the height.

Short answer

The area under the parabolic arch is found to be the definite integral of the function from -3 to 2, the height is the y-coordinate of the maximum point of the parabola within the interval [-3,2] and the verification of Archimedes' principle can be confirmed by checking if the calculated area equals two-thirds the base times the height.

Step-by-step solution

Find the Area Under the Parabolic Arch

With the given inequality, -3 ≤ x ≤ 2, the area A under the parabolic arch y = 6 - x - x^2 can be found by integrating the function from -3 to 2. In math terms, we calculate the definite integral \(\int_{-3}^{2}(6-x-x^2) dx \).

Find the Height of the Arch

The maximum point of the parabola within the interval [-3,2] will give the height of the arch. The x-coordinate of the maximum point is given by the formula -b/2a where 'a' and 'b' are the coefficients of \(x^2\) and 'x' respectively for the function y = 6 - x - x^2. The y-coordinate (the height) is the value of the function at this x-coordinate.

Verify Archimedes' Principle for Parabolic Arch

Finally, using the equation A = 2/3 * base * height, with base = 2 - (-3) = 5, and height as found in Step 2, it should be shown whether the calculated area A from Step 1 equals this value. The accuracy of Archimedes' principle can be verified by comparing both the results.