In the next step of the derivation, we passed a horizontal plane through the sphere b up from the center. We let the radius of the cross section be x and formed a right triangle with hypotenuse r, as shown below.

The cross-sectional area of the shaded circle is πx2. Using the Pythagorean theorem for the right triangle, we get x2 + b2 = r2. Now solve for x2 and substitute it into the area expression.

What is the result?

π(r2 – b2)
π(r2 + b2)
π(b2 – r2)
2π(b2 + r2)

According to a study conducted in 2015, 18% of shoppers said that they prefer to buy generic instead of name-brand products. suppose that in a recent sample of 1500 shoppers, 315 stated that they prefer to buy generic instead of name-brand products. at a 5% significance level, can you conclude that the proportion of all shoppers who currently prefer to buy generic instead of name-brand products is higher than .18? use both the p-value and the critical-value approaches.