Questions: For the following question, fill in the blank to complete the statement. (There may be more than one correct answer.) a) If the volume of a sphere is doubled, its radius increases by a factor of . b) If the radius of a sphere is doubled, its volume increases by a factor of . c) The solution sets to f(x) ≤ c and x f(x) ≤ c are (the same/different).

For the following question, fill in the blank to complete the statement. (There may be more than one correct answer.)

a) If the volume of a sphere is doubled, its radius increases by a factor of .

b) If the radius of a sphere is doubled, its volume increases by a factor of .

c) The solution sets to f(x) ≤ c and x f(x) ≤ c are (the same/different).

Transcript text: For the following question, fill in the blank to complete the statement. (There may be more than one correct answer.) a) If the volume of a sphere is doubled, its radius increases by a factor of ______. b) If the radius of a sphere is doubled, its volume increases by a factor of ______. c) The solution sets to f(x) ≤ c and {x | f(x) ≤ c} are (the same/different).

Solution

Solution Steps

Step 1: Solve part (a) - Volume of a sphere doubled

The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] If the volume is doubled, the new volume \( V' \) is: \[ V' = 2V = 2 \left( \frac{4}{3} \pi r^3 \right) = \frac{8}{3} \pi r^3 \] Let the new radius be \( r' \). Then: \[ \frac{4}{3} \pi (r')^3 = \frac{8}{3} \pi r^3 \] Divide both sides by \( \frac{4}{3} \pi \): \[ (r')^3 = 2r^3 \] Take the cube root of both sides: \[ r' = \sqrt[3]{2} \, r \] Thus, the radius increases by a factor of \( \sqrt[3]{2} \).

Step 2: Solve part (b) - Radius of a sphere doubled

If the radius is doubled, the new radius \( r' \) is: \[ r' = 2r \] The new volume \( V' \) is: \[ V' = \frac{4}{3} \pi (r')^3 = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = 8 \left( \frac{4}{3} \pi r^3 \right) = 8V \] Thus, the volume increases by a factor of \( 8 \).

Step 3: Solve part (c) - Solution sets comparison

The solution set to \( f(x) \leq c \) is the set of all \( x \) such that \( f(x) \leq c \). This is exactly the same as the set \( \{x \mid f(x) \leq c\} \). Therefore, the solution sets are the same.