UpStudy Free Solution:

To find the depth (run) of the staircase given the height (rise) and the ratio of rise to run, we can follow these steps:

1. Convert the rise and run from mixed numbers to improper fractions:

Rise: \(6 \frac { 3} { 4} \) inches

\[6 \frac { 3} { 4} = 6 + \frac { 3} { 4} = \frac { 24} { 4} + \frac { 3} { 4} = \frac { 27} { 4} \space \text { inches} \]

Run: \(7 \frac { 3} { 4} \) inches

\[7 \frac { 3} { 4} = 7 + \frac { 3} { 4} = \frac { 28} { 4} + \frac { 3} { 4} = \frac { 31} { 4} \space \text { inches} \]

2. Find the ratio of rise to run:

\[\text { Ratio} = \frac { \frac { 27} { 4} \space \text { inches} } { \frac { 31} { 4} \space \text { inches} } = \frac { 27} { 31} \]

3. Convert the total rise of the staircase from feet to inches:

Total rise: \(17\) feet

\[17 \space \text { feet} \times 12 \space \text { inches/foot} = 204 \space \text { inches} \]

4. Calculate the total run using the ratio:

\[\frac { \text { Total rise} } { \text { Total run} } = \frac { 27} { 31} \]

Let \(\text { Total run} \) be \(x\) inches:

\[\frac { 204} { x} = \frac { 27} { 31} \]

Solving for \(x\):

\[x = \frac { 204 \times 31} { 27} \]

\[x = \frac { 6324} { 27} \]

\[x \approx 234.2 \space \text { inches} \]

5. Convert the total run from inches to feet:

\[\text { Total run in feet} = \frac { 234.2 \space \text { inches} } { 12 \space \text { inches/foot} } \approx 19.5 \space \text { feet} \]

Final Answer

The staircase will need to be approximately \(19.5\) feet deep.

So, the depth of the staircase is approximately \(19.5\) feet.

Supplemental Knowledge

Ratios and Proportions in Staircase Construction

When constructing staircases, builders often follow specific guidelines for the rise (vertical height of each step) and run (horizontal depth of each step). The ratio of rise to run must be consistent to ensure safety and comfort. This ratio can be used to determine the total horizontal distance (run) required for a given vertical height (rise).

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