Question

1. what is the distance between the bank and hirotos house on the map? write the distance to the nearest quarter mile. (each unit represents 1 mile.)
2. find the distance between (-3, -6) and (-3, 1).
3. the graph shows the locations of point c and point d. point e is graphed at (n, 2). the distance from point d to point e is equal to the distance from point d to point c. what is the distance from point d to point e? what is the value of n?
4. select the two ordered pairs that are 5.5 units apart. (-3, -3.5) (-1.5, 2) (-1.5, 2.5) (5, -3.5) (-1.5, -3.5)
5. the map at the right shows the beach area on a coordinate plane. what is the total distance in miles from the car to the cabana and then from the cabana to the beach?

Explanation:

Step1: <Find coordinates>

Let the bank's coordinates be \((-8,5)\) and Hiroto's house coordinates be \((2,6)\).
Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Here \(x_1=-8,y_1 = 5,x_2=2,y_2 = 6\).

Step2: <Calculate \(x\) and \(y\) differences>

\(x_2 - x_1=2-(-8)=10\), \(y_2 - y_1=6 - 5 = 1\).

Step3: <Apply distance formula>

\(d=\sqrt{(10)^2+(1)^2}=\sqrt{100 + 1}=\sqrt{101}\approx10.05\) miles.
Rounding to the nearest quarter - mile (\(0.25\) mile), \(10.05\div0.25 = 40.2\), so \(d\approx10.0\) miles.

Step4: <For the second problem>

For points \((-3,-6)\) and \((-3,1)\), since \(x\) - coordinates are the same (\(x=-3\)), use the formula \(d=\vert y_2 - y_1\vert\).
\(d=\vert1-(-6)\vert=\vert1 + 6\vert=7\) units.

Step5: <For the third problem>

Let \(C=(0,-3)\) and \(D=(n,-2)\). The distance from \(D\) to \(C\) is \(d_{DC}=\sqrt{(n - 0)^2+(-2+3)^2}=\sqrt{n^{2}+1}\).
The distance from \(D=(n,-2)\) to \(E=(n,2)\) is \(d_{DE}=\vert2-(-2)\vert = 4\).
Since \(d_{DC}=d_{DE}\), \(\sqrt{n^{2}+1}=4\).
Squaring both sides: \(n^{2}+1 = 16\), \(n^{2}=15\), \(n=\pm\sqrt{15}\approx\pm3.87\).

Step6: <For the fourth problem>

Check the distance between pairs using \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) or \(\vert x_2 - x_1\vert\) (if \(y\) - coordinates are same) or \(\vert y_2 - y_1\vert\) (if \(x\) - coordinates are same).
For \((-1.5,2)\) and \((-1.5,-3.5)\): \(d=\vert2-(-3.5)\vert=\vert2 + 3.5\vert=5.5\) units.

Step7: <For the fifth problem>

Let car's coordinates be \((-4,5)\), cabana's coordinates be \((-3,0)\), beach's coordinates be \((2,1)\).
Distance from car to cabana: \(d_1=\sqrt{(-3 + 4)^2+(0 - 5)^2}=\sqrt{1+25}=\sqrt{26}\approx5.1\) miles.
Distance from cabana to beach: \(d_2=\sqrt{(2 + 3)^2+(1 - 0)^2}=\sqrt{25 + 1}=\sqrt{26}\approx5.1\) miles.
Total distance \(d=d_1 + d_2\approx5.1+5.1 = 10.2\) miles.

Answer:

1. Approximately \(10.0\) miles.
2. \(7\) units.
3. \(n=\pm\sqrt{15}\approx\pm3.87\).
4. \((-1.5,2)\) and \((-1.5,-3.5)\).
5. Approximately \(10.2\) miles.