Question

A bolt extends through three fourths
​-inch-thick ​plywood, two washers that are each three sixteenths
inch​ thick, and a nut that is one sixteenth
inch thick. The main body of the bolt must be one half
inch longer than the sum of the thicknesses of​ plywood, washers, and nut. What is the minimum length of the​ bolt?
Question content area bottom
Part 1
The minimum length of the bolt is
  
enter your response here inch.

Answer 1

To find the minimum length of the bolt, we first need to calculate the total thickness of the plywood, washers, and nut.

1. **Thickness of plywood**: \( \frac{3}{4} \) inch
2. **Thickness of two washers**: Each washer is \( \frac{3}{16} \) inch. Since there are two washers, the total thickness is:
\[
2 \times \frac{3}{16} = \frac{6}{16} = \frac{3}{8} \text{ inch}
\]
3. **Thickness of nut**: \( \frac{1}{16} \) inch

Now, sum these thicknesses:
\[
\text{Total thickness} = \frac{3}{4} + \frac{3}{8} + \frac{1}{16}
\]

To add these fractions, we first convert them to a common denominator. The least common multiple (LCM) of 4, 8, and 16 is 16.

Convert each fraction to the common denominator of 16:
- \( \frac{3}{4} = \frac{12}{16} \)
- \( \frac{3}{8} = \frac{6}{16} \)
- \( \frac{1}{16} = \frac{1}{16} \)

Now, add these fractions:
\[
\text{Total thickness} = \frac{12}{16} + \frac{6}{16} + \frac{1}{16} = \frac{19}{16} \text{ inch}
\]

Next, we need the main body of the bolt to be \( \frac{1}{2} \) inch longer than the total thickness:
- \( \frac{1}{2} \) inch in terms of sixteenths is \( \frac{8}{16} \) inch.

Thus, the length of the bolt should be:
\[
\text{Minimum length of the bolt} = \text{Total thickness} + \frac{1}{2} = \frac{19}{16} + \frac{8}{16} = \frac{27}{16} \text{ inch}
\]

Finally, we can convert \( \frac{27}{16} \) inch to a mixed number:
\[
\frac{27}{16} = 1 \frac{11}{16} \text{ inch}
\]

Therefore, the minimum length of the bolt is:
\[
\boxed{1 \frac{11}{16}} \text{ inch}
\]