The amount of carbon dioxide (in pounds per 15,000 cubic miles) released by a certain SUV depends on its fuel efficiency according to the formula

W = 33x2 − 2145x + 43,000 (12 ≤ x ≤ 33)
where x is the fuel efficiency of an SUV in miles per gallon. According to the model, what is the fuel efficiency of the SUV with the least carbon dioxide pollution?

x = ___ mi per gallon

I'm not sure what formula to use to solve this equation. As well as how to get the answer. How should I approach this problem?

Thank you

To find the fuel efficiency of the SUV with the least carbon dioxide pollution, we need to minimize the function W(x) = 33x^2 - 2145x + 43,000 within the given range of 12 ≤ x ≤ 33.

To do this, we can use calculus and find the derivative of the function W(x) with respect to x. We then set the derivative equal to zero and solve for x to find the critical points.

Let's go through the steps to solve this problem:

Step 1: Find the derivative of the function W(x)
The derivative of W(x) is given by:

dW/dx = 66x - 2145

Step 2: Set the derivative equal to zero and solve for x
Setting dW/dx = 0, we have:

66x - 2145 = 0

Solving for x:

66x = 2145
x = 2145/66
x ≈ 32.5

Step 3: Check if the critical point is within the given range
Since the given range is 12 ≤ x ≤ 33, we need to check if x = 32.5 falls within this range. In this case, it does, so we can consider it as a potential solution.

Step 4: Check the endpoints of the given range
We also need to check the endpoints of the given range (x = 12 and x = 33) to see if they could potentially be the solution. To do this, we evaluate the function W(x) at these endpoints.

W(12) = 33(12)^2 - 2145(12) + 43,000 ≈ 25,800
W(33) = 33(33)^2 - 2145(33) + 43,000 ≈ 21,600

Step 5: Identify the minimum value of W(x)
Comparing the values of W(x) at the critical point and the endpoints within the given range, we see that W(33) = 21,600 is the smallest value.

Therefore, the fuel efficiency of the SUV with the least carbon dioxide pollution is approximately x = 33 miles per gallon.

Note: These calculations assume the function W(x) accurately models the relationship between fuel efficiency and carbon dioxide pollution. However, it should be noted that the real-world relationship may be more complex and involve additional factors.

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