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The problem involves the fuel consumption model of a car given by the equation:

\[
y = 55 - 0.015(x - 50)^2
\]

where:
- \( y \) is the fuel consumption (in miles per gallon),
- \( x \) is the speed of the car (in miles per hour).

Let’s address the parts one by one.

### (a) Calculate the fuel consumption when \( x = 70 \) miles per hour.
We need to substitute \( x = 70 \) into the equation:

\[
y = 55 - 0.015(70 - 50)^2
\]

\[
y = 55 - 0.015(20)^2
\]

\[
y = 55 - 0.015(400)
\]

\[
y = 55 - 6 = 49
\]

So, the fuel consumption at 70 miles per hour is **49 miles per gallon**.

### (b) Deduce the speed at which the car has the greatest fuel efficiency.
The model is quadratic in form. The maximum fuel efficiency corresponds to the vertex of the parabola, since the quadratic term has a negative coefficient (indicating a downward parabola).

The vertex of a parabola \( y = a(x - h)^2 + k \) occurs at \( x = h \). Here, \( (x - 50)^2 \) suggests the vertex occurs at:

\[
x = 50 \text{ miles per hour.}
\]

Thus, the car has the greatest fuel efficiency at **50 miles per hour**.

### (c) State, giving reasons, the limitation on the values of \( x \).
The function is a downward parabola, so the fuel consumption \( y \) decreases as \( x \) approaches 50, but increases again as \( x \) moves further from 50 in either direction.

#### Limitation:
- Since the fuel consumption \( y \) must be positive (i.e., the car must consume fuel), the value of \( x \) should be limited so that \( y \geq 0 \). To find this, set \( y = 0 \):

\[
55 - 0.015(x - 50)^2 = 0
\]

\[
0.015(x - 50)^2 = 55
\]

\[
(x - 50)^2 = \frac{55}{0.015} = 3666.67
\]

\[
x - 50 = \pm \sqrt{3666.67} \approx \pm 60.53
\]

\[
x = 50 \pm 60.53
\]

So, \( x \approx 110.53 \) or \( x \approx -10.53 \). Since speeds cannot be negative, the range of \( x \) is:

\[
0 \leq x \leq 110.53
\]

Thus, the car's speed is limited to between **0 and approximately 110.53 miles per hour**, beyond which the fuel consumption model is not valid.

---

Would you like further details or have any other questions?

Here are 5 related questions you can explore:
1. How would the fuel consumption change at \( x = 30 \) miles per hour?
2. What is the fuel consumption at the maximum efficiency of 50 miles per hour?
3. How does the coefficient \( -0.015 \) affect the shape of the fuel consumption curve?
4. What happens to the fuel consumption as \( x \) approaches the upper limit of the speed?
5. Can we adjust the model to allow higher speeds while keeping fuel consumption valid?

**Tip:** For quadratic models, the vertex always gives the point of maximum or minimum depending on the sign of the squared term.

This model can be used to predict the fuel consumption of a car, y miles per gallon, for a car travelling at x miles per hour. The equation is y = 55 - 0.015(x - 50)^2. (a) Calculate the fuel consumption in miles per gallon, of a car travelling at 70 miles per hour. (b) Deduce the speed at which the car has the greatest fuel efficiency. (c) State, giving reasons, the limitation on the values of x.

Analyze the quadratic model y = 55 - 0.015(x - 50)^2 to predict fuel consumption at different speeds. Learn how to calculate the fuel consumption at 70 mph, find the speed with maximum fuel efficiency, and determine the limitations on the speed variable x. Suitable for high school students in Grades 10-12.

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