Question

Theo, Nancy, and Mary each drove separately to meet at their grandmother’s house for dinner. Theo drove 50 miles, Nancy drove √2,840 miles, and Mary’s route is shown below.

A right triangle with side lengths 48 miles and 13 miles. The hypotenuse is labeled Mary's Route.

Approximately how many miles did Mary drive to her grandmother’s house?



Order the distances that each family member drove to their grandmother’s house from least to greatest.

Answer 1

To find the distance Mary drove, we need to calculate the hypotenuse of the right triangle with side lengths 48 miles and 13 miles. We can use the Pythagorean theorem, which states that in a right triangle:

\[
c = \sqrt{a^2 + b^2}
\]

where \( c \) is the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.

Here, we have:
- \( a = 48 \) miles
- \( b = 13 \) miles

Now, plugging these values into the formula:

\[
c = \sqrt{48^2 + 13^2}
\]

Calculating the squares of the sides:

\[
48^2 = 2304
\]
\[
13^2 = 169
\]

Now, we add these together:

\[
c = \sqrt{2304 + 169} = \sqrt{2473}
\]

To find \( \sqrt{2473} \), we can approximate it. Since \( 50^2 = 2500 \) and \( 49^2 = 2401 \), we know that:

\[
49 < \sqrt{2473} < 50
\]

Calculating \( \sqrt{2473} \) gives approximately:

\[
\sqrt{2473} \approx 49.73 \text{ miles}
\]

Now we summarize the distances each person drove:

- Theo: 50 miles
- Nancy: \( \sqrt{2840} \)

Calculating \( \sqrt{2840} \):

\[
\sqrt{2840} \approx \sqrt{2800} \approx \sqrt{400 \times 7} = 20\sqrt{7}
\]

Since \( \sqrt{7} \approx 2.64575 \):

\[
20\sqrt{7} \approx 20 \times 2.64575 \approx 52.915 \text{ miles}
\]

Now, we have the following distances:

- Theo: 50 miles
- Nancy: approximately 52.915 miles
- Mary: approximately 49.73 miles

Ordering these distances from least to greatest:

1. Mary: approximately 49.73 miles
2. Theo: 50 miles
3. Nancy: approximately 52.915 miles

Thus, the final order of distances from least to greatest is:

1. Mary: 49.73 miles
2. Theo: 50 miles
3. Nancy: 52.915 miles