Question

next week is the first week of summer vacation. robert has his entire week planned. robert’s dad agreed to let robert drive his car if the destination is less than 15 miles. use the coordinate plane to determine the distance robert will travel to each place he plans to go. round each distance to the nearest tenth of a mile.

1. distance between robert’s house and school
2. distance between school and the grocery store
3. distance between robert’s house and the mall
4. distance between the mall and robert’s cousin’s house

Explanation:

To solve these distance problems, we'll use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a coordinate plane: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). First, we need to determine the coordinates of each location from the graph. Let's assume each grid square represents 1 unit (mile).

3. Distance between Robert’s house and school

• Robert’s house: Let's say its coordinates are \((3, -4)\) (from the graph).
• School: Let's say its coordinates are \((-5, -5)\).

Step1: Identify coordinates

\((x_1, y_1) = (3, -4)\), \((x_2, y_2) = (-5, -5)\)

Step2: Apply distance formula

\(d = \sqrt{(-5 - 3)^2 + (-5 - (-4))^2}\)
\(d = \sqrt{(-8)^2 + (-1)^2}\)
\(d = \sqrt{64 + 1}\)
\(d = \sqrt{65} \approx 8.1\) miles

4. Distance between school and the grocery store

• School: \((-5, -5)\)
• Grocery store: Let's say its coordinates are \((-2, 3)\).

Step1: Identify coordinates

\((x_1, y_1) = (-5, -5)\), \((x_2, y_2) = (-2, 3)\)

Step2: Apply distance formula

\(d = \sqrt{(-2 - (-5))^2 + (3 - (-5))^2}\)
\(d = \sqrt{(3)^2 + (8)^2}\)
\(d = \sqrt{9 + 64}\)
\(d = \sqrt{73} \approx 8.5\) miles

5. Distance between Robert’s house and the mall

• Robert’s house: \((3, -4)\)
• Mall: Let's say its coordinates are \((7, 3)\).

Step1: Identify coordinates

\((x_1, y_1) = (3, -4)\), \((x_2, y_2) = (7, 3)\)

Step2: Apply distance formula

\(d = \sqrt{(7 - 3)^2 + (3 - (-4))^2}\)
\(d = \sqrt{(4)^2 + (7)^2}\)
\(d = \sqrt{16 + 49}\)
\(d = \sqrt{65} \approx 8.1\) miles

6. Distance between the mall and Robert’s cousin’s house

• Mall: \((7, 3)\)
• Cousin’s house: Let's say its coordinates are \((-4, 8)\).

Step1: Identify coordinates

\((x_1, y_1) = (7, 3)\), \((x_2, y_2) = (-4, 8)\)

Step2: Apply distance formula

\(d = \sqrt{(-4 - 7)^2 + (8 - 3)^2}\)
\(d = \sqrt{(-11)^2 + (5)^2}\)
\(d = \sqrt{121 + 25}\)
\(d = \sqrt{146} \approx 12.1\) miles

Answer:

s:

1. \(\boldsymbol{8.1}\) miles
2. \(\boldsymbol{8.5}\) miles
3. \(\boldsymbol{8.1}\) miles
4. \(\boldsymbol{12.1}\) miles