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Question

Part 1
The formula Upper C equals 3.14 d can be used to approximate the circumference of a circle given its diameter. Company A manufactures and sells a certain washer with an outside circumference of 11 centimeters. The company has decided that a washer whose actual circumference is in the interval 10.9less than or equalsCless than or equals11.1 centimeters is acceptable. Use a compound inequality and find the corresponding interval for diameters of these washers.

Answer 1

To find the corresponding interval for the diameters of the washers, we start with the formula for circumference, which is given by:

\[ C = 3.14d \]

where \( C \) is the circumference and \( d \) is the diameter.

We know that the circumference \( C \) of the washers is acceptable in the interval:

\[ 10.9 \leq C \leq 11.1 \]

Next, we'll substitute the formula for circumference into the inequality:

1. For the lower bound:

\[ 10.9 \leq 3.14d \]

To isolate \( d \), divide both sides by 3.14:

\[ \frac{10.9}{3.14} \leq d \]

Calculating \( \frac{10.9}{3.14} \):

\[ \frac{10.9}{3.14} \approx 3.4701 \]

So, the lower bound for \( d \) is approximately:

\[ d \geq 3.4701 \]

2. For the upper bound:

\[ 3.14d \leq 11.1 \]

Again, isolate \( d \) by dividing both sides by 3.14:

\[ d \leq \frac{11.1}{3.14} \]

Calculating \( \frac{11.1}{3.14} \):

\[ \frac{11.1}{3.14} \approx 3.5375 \]

So, the upper bound for \( d \) is approximately:

\[ d \leq 3.5375 \]

Now we can combine these results into a compound inequality that describes the acceptable interval for the diameters of the washers:

\[ 3.4701 \leq d \leq 3.5375 \]

Therefore, the corresponding interval for the diameters of the washers is approximately:

\( [3.4701, 3.5375] \) centimeters.