The tolerance in inches for the diameter of a piston is described by the absolute value inequality |d-5| , or equal to 0.01. What is the desired diameter of the piston? By how much can the actual diamer of the piston vary from the desired diameter?
|d-5} < .01
d-5 <.01 and d-5>-.01
d < 5.01 and d > 4.99
the desired diameter would be when |d-5| = 0
or
d = 5
Well, well, well, looks like we've got some serious piston diameter business going on here! Let me put on my comedy hat and give you some answers.
The absolute value inequality |d-5| ≤ 0.01 means that the difference between the desired diameter (d) and 5 should be less than or equal to 0.01.
To find the desired diameter, we need to solve the equation |d-5| = 0.01. Since we have an absolute value, we need to consider two cases:
Case 1: d - 5 = 0.01
In this case, we add 5 to both sides of the equation to isolate d:
d = 5 + 0.01
d = 5.01
Case 2: -(d - 5) = 0.01
Here, we distribute the negative sign inside the absolute value:
-d + 5 = 0.01
Subtract 5 from both sides:
-d = -4.99
Multiply by -1 to isolate d:
d = 4.99
So, the desired diameter of the piston can be either 5.01 inches or 4.99 inches (I hope it doesn't suffer from an identity crisis!).
Now, as for how much the actual diameter can vary from the desired diameter, we simply need to calculate the difference between the maximum and minimum values:
Difference = 5.01 - 4.99
Difference = 0.02 inches.
Hence, the actual diameter of the piston can vary by 0.02 inches from the desired diameter. That's not too shabby in the piston world. Now let's get back to clowning around, shall we?
To find the desired diameter of the piston, we need to solve the absolute value inequality |d-5| ≤ 0.01.
First, we set up the two inequalities:
d - 5 ≤ 0.01 (since |d-5| = d-5 when d-5 is positive)
-(d - 5) ≤ 0.01 (since |d-5| = -(d-5) when d-5 is negative)
Simplifying each inequality, we have:
d ≤ 0.01 + 5
-d + 5 ≤ 0.01
Simplifying further, we get:
d ≤ 5.01
-d ≤ -4.99
Now, we solve for d in each inequality:
For the first inequality:
d ≤ 5.01
For the second inequality, we have to multiply both sides by -1, but when we do that, we need to reverse the direction of the inequality:
d ≥ 4.99
Therefore, the desired diameter of the piston falls between 4.99 and 5.01 inches.
To find the amount by which the actual diameter can vary from the desired diameter, we can use the difference between the upper and lower limits:
Upper Limit - Lower Limit = 5.01 - 4.99 = 0.02 inches
Therefore, the actual diameter of the piston can vary by up to 0.02 inches from the desired diameter.
To find the desired diameter and the acceptable range of variation for the actual diameter of the piston, let's analyze the given absolute value inequality:
|d - 5| ≤ 0.01
This inequality states that the absolute value of the difference between the diameter of the piston (d) and 5 is less than or equal to 0.01.
To determine the desired diameter, we need to find the value of d that satisfies this inequality.
To solve the absolute value inequality |d - 5| ≤ 0.01, we have two cases:
Case 1: (d - 5) ≤ 0.01
In this case, the absolute value is unnecessary because the expression within the absolute value brackets is already less than or equal to 0.01. Solving for d:
d - 5 ≤ 0.01
d ≤ 0.01 + 5
d ≤ 5.01
Case 2: -(d - 5) ≤ 0.01
In this case, we include the negative sign in front of the absolute value brackets because we need to consider the negative difference as well. Solving for d:
-d + 5 ≤ 0.01
5 - d ≤ 0.01
5 - 0.01 ≤ d
4.99 ≤ d
By analyzing the values of d obtained from each case, we can conclude that the desired diameter of the piston can range from 4.99 to 5.01 inches.
To determine the acceptable variation, we calculate the difference between the maximum and minimum values:
Maximum diameter - Minimum diameter = 5.01 - 4.99 = 0.02 inches.
Therefore, the actual diameter of the piston can vary by a maximum of 0.02 inches from the desired diameter.