Section 5.5: Applications & Review

This section applies radical functions and rational exponents to real-world problems and provides a review of key concepts covered in the chapter.

Example 1: Solving a Real-World Problem

The height of a ball dropped from a 100 m building is modeled by \( h(t) = \sqrt{100 - 9.8t^2} \). Find the height after 2 seconds.

Step 1: Substitute t = 2 → \( h(2) = \sqrt{100 - 9.8(2)^2} \)

Step 2: Compute inside the square root → \( 100 - 39.2 = 60.8 \)

Step 3: Take square root → \( h(2) \approx 7.8 \) m

Simplify \( (27x^6)^{2/3} \) in an engineering formula.

Step 1: Rewrite with radicals → \( \sqrt[3]{(27x^6)^2} \)

Step 2: Simplify inside → \( \sqrt[3]{729 x^{12}} \)

Step 3: Cube root → \( 9 x^4 \)

Evaluate \( \sqrt{81} + \sqrt[3]{64} \)
Solve \( x^{3/2} = 27 \) for x
Graph \( f(x) = \sqrt{x-1} + 2 \)
Simplify \( (16x^8)^{3/4} \)
Application: The radius of a sphere grows as \( r(t) = \sqrt{t+1} \). Find r at t=8.