[Technology Exercise] Grinding engine cylinders Before contracting to grind engine cylinders to a cross-sectional area of 9in², you need to know how much deviation from the ideal cylinder diameter of c = 3.385in. you can allow and still have the area come within 0.01in² of the required 9in². To find out, you let A=π(x/2)² and look for the largest interval in which you must hold x to make |A − 9| ≤ 0.01. What interval do you find?

Verified step by step guidance

1. Start by understanding the problem: You need to find the interval for the diameter x such that the area A of the cylinder's cross-section is within 0.01 in² of 9 in². The formula for the area A is given by A = π(x/2)².
2. Set up the inequality |A - 9| ≤ 0.01. Substitute the expression for A into the inequality: |π(x/2)² - 9| ≤ 0.01.
3. Solve the inequality: First, express the inequality without the absolute value by considering two cases: π(x/2)² - 9 ≤ 0.01 and π(x/2)² - 9 ≥ -0.01.
4. For the first case, π(x/2)² - 9 ≤ 0.01, solve for x by isolating x: π(x/2)² ≤ 9.01. Then, divide both sides by π and take the square root to find x.
5. For the second case, π(x/2)² - 9 ≥ -0.01, solve for x similarly: π(x/2)² ≥ 8.99. Divide both sides by π and take the square root to find x. Combine the results from both cases to determine the interval for x.

Key Concepts

Area of a Circle

The area of a circle is calculated using the formula A = π(r²), where r is the radius. In this context, the radius is half of the diameter (x/2). Understanding this formula is crucial for determining how changes in the diameter affect the area, which is central to solving the problem.

Deviation and Tolerance

Deviation refers to the difference between a measured value and a standard or ideal value. In this problem, the tolerance is set at 0.01 in², meaning the area must remain within 0.01 in² of the target area of 9 in². This concept is essential for establishing the acceptable range of values for the diameter.

Inequalities in Calculus

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. In this case, the inequality |A - 9| ≤ 0.01 is used to find the range of diameters that keep the area within the specified limits. Understanding how to manipulate and solve inequalities is key to finding the required interval for x.