Question

The length of the curve \[ y = \frac{3}{4} x^{4/3} - \frac{3}{8} x^{2/3} + 7 \] from \( x = 1 \) to \( x = 8 \) equals

• A. \( \frac{99}{8} \)
• B. \( \frac{117}{8} \)
• C. \( \frac{99}{4} \)
• D. \( \frac{49}{8} \)

Hint:

To calculate the length of a curve, differentiate the function, square the derivative, add 1, and integrate over the desired interval.

Answer 1

The Correct Option is A

Step 1: Formula for the length of a curve.
The formula for the length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \) is: \[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx \] We will first find \( \frac{dy}{dx} \) and then integrate. Step 2: Differentiate the function.
The derivative of \( y = \frac{3}{4} x^{4/3} - \frac{3}{8} x^{2/3} + 7 \) is: \[ \frac{dy}{dx} = \frac{3}{4} \cdot \frac{4}{3} x^{1/3} - \frac{3}{8} \cdot \frac{2}{3} x^{-1/3} = \frac{3}{3} x^{1/3} - \frac{1}{4} x^{-1/3} \] Simplify to: \[ \frac{dy}{dx} = x^{1/3} - \frac{1}{4} x^{-1/3} \] Step 3: Calculate the length.
Substitute into the length formula and calculate the integral from 1 to 8. Step 4: Conclusion.
The correct answer is (B) \( \frac{117}{8} \).