Given:
- Function: [tex]Y(x) = 0.03x^3 - 0.002x^2 - 0.0005x + 20[/tex] feet
- At [tex]x = 8[/tex] feet, the speed at this point is [tex]35 \, \text{ft/s}[/tex]
- Speed changes at a rate of [tex]-2 \, \text{ft/s}^2[/tex]
- Coefficient of friction: [tex]0.35[/tex]

What is the value of the function [tex]Y(x)[/tex] at [tex]x = 8[/tex] feet?

A. 35
B. 20
C. 1.64
D. 0.3

To find the value of Y(x) at x = 8 feet, plug x = 8 into the polynomial Y(x) = 0.03x³ - 0.002x² - 0.0005x + 20, resulting in the value of approximately 35.228. There might be a rounding or typographical error as this does not match the given options exactly.

This correct answer is a)

Explanation:

The value of the function Y(x) at x = 8 feet can be directly evaluated from the given function Y(x) = 0.03x³ - 0.002x² - 0.0005x + 20. By substituting x = 8 into the equation:

Y(8) = 0.03(8)³ - 0.002(8)² - 0.0005(8) + 20

Calculate each term:

0.03(8)³ = 0.03(512) = 15.36
-0.002(8)² = -0.002(64) = -0.128
-0.0005(8) = -0.004
+ 20

Summing those up gives:

Y(8) = 15.36 - 0.128 - 0.004 + 20 = 35.228

However, this value does not match directly with any of the presented multiple-choice answers, indicating a possible error in the question or in the interpretation of the information provided. The value that most closely resembles the correct calculation is option (a) 35, but one must assume a rounding or typographical error to choose this.

This correct answer is a)