The function F(x)= -0.005x^2 +0.8x+12  models the relationship between a certain vehicle’s speed and fuel economy, where F(x) is the fuel economy in kilometres per litre and xis the speed of the vehicle in kilometres per hour. Determine the rate of change in fuel economy for 10 km/h intervals in speed, and use your results to determine the speed that gives the best fuel economy


To determine the rate of change in fuel economy for 10 km/h intervals in speed, we need to find the derivative of the function \( F(x) \). The derivative, \( F'(x) \), gives us the rate of change of fuel economy with respect to speed.

1. **Find the derivative \( F'(x) \):**

\[
F(x) = -0.005x^2 + 0.8x + 12
\]

Taking the derivative with respect to \( x \):

\[
F'(x) = -0.01x + 0.8
\]

2. **Calculate the rate of change for 10 km/h intervals:**

The rate of change between two speeds \( x_1 \) and \( x_2 \) (where \( x_2 = x_1 + 10 \)) can be approximated using the derivative \( F'(x) \). For example:

- At \( x = 10 \):
  \[
  F'(10) = -0.01(10) + 0.8 = 0.7 \, \text{km/L per km/h}
  \]

- At \( x = 20 \):
  \[
  F'(20) = -0.01(20) + 0.8 = 0.6 \, \text{km/L per km/h}
  \]

- At \( x = 30 \):
  \[
  F'(30) = -0.01(30) + 0.8 = 0.5 \, \text{km/L per km/h}
  \]

- At \( x = 40 \):
  \[
  F'(40) = -0.01(40) + 0.8 = 0.4 \, \text{km/L per km/h}
  \]

- At \( x = 50 \):
  \[
  F'(50) = -0.01(50) + 0.8 = 0.3 \, \text{km/L per km/h}
  \]

3. **Determine the speed that gives the best fuel economy:**

The best fuel economy occurs at the maximum point of the function \( F(x) \). This happens where the derivative \( F'(x) = 0 \).

\[
-0.01x + 0.8 = 0
\]
\[
x = 80 \, \text{km/h}
\]

So, the speed that gives the best fuel economy is 80 km/h.

**Summary:**
- The derivative \( F'(x) = -0.01x + 0.8 \) gives the rate of change of fuel economy.
- The speed that provides the best fuel economy is 80 km/h.

Do you want more details on any part, or have questions about this? Here are some related questions:

1. How does the rate of change in fuel economy vary at different speeds?
2. What does a positive or negative rate of change in \( F'(x) \) indicate about fuel economy?
3. Can you calculate the fuel economy at different specific speeds using \( F(x) \)?
4. How can we find the maximum or minimum points of a quadratic function like \( F(x) \)?
5. Why is the derivative useful in understanding changes in fuel economy?

**Tip:** The maximum or minimum point of a quadratic function \( ax^2 + bx + c \) is always at \( x = -\frac{b}{2a} \).

The function F(x) = -0.005x^2 + 0.8x + 12 models the relationship between a certain vehicle’s speed and fuel economy, where F(x) is the fuel economy in kilometres per litre and x is the speed of the vehicle in kilometres per hour. Determine the rate of change in fuel economy for 10 km/h intervals in speed, and use your results to determine the speed that gives the best fuel economy.

Explore how to use derivatives to find the rate of change in fuel economy for a vehicle's speed using the function F(x) = -0.005x^2 + 0.8x + 12. Learn to determine the optimal speed for the best fuel economy, featuring calculus concepts and quadratic optimization. Suitable for Grades 11-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation