Question

how far is it from the restaurant to the bicycle shop? submit units hotel science lab restaurant bicycle shop harbor radio tower

Explanation:

Step1: Determine the coordinates

The restaurant is at \((-1,-1)\) and the bicycle shop is at \((3,-3)\).

Step2: Use the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1=-1,y_1 = -1,x_2=3,y_2=-3\).

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Alternatively, using the grid - count (horizontal and vertical distance):
The horizontal distance between \(x=-1\) and \(x = 3\) is \(|3-(-1)|=4\) units. The vertical distance between \(y=-1\) and \(y=-3\) is \(|-3-(-1)|=2\) units. Then, using the Pythagorean theorem \(d=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\approx4.47\). But if we count the number of units by moving from the restaurant to the bicycle shop (right 4 units and down 2 units and then using the Pythagorean theorem concept for the shortest path), we can also think in terms of the difference in coordinates.

Another way (if we consider the movement along the axes):
The change in the \(x\) - coordinate: \(\Delta x=3-(-1)=4\), the change in the \(y\) - coordinate: \(\Delta y=-3-(-1)=-2\). The distance \(d=\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}=\sqrt{4^{2}+(-2)^{2}}=\sqrt{16 + 4}=\sqrt{20}\approx4.47\). But if we assume that we are just asked for the number of units by moving along the grid (not the straight - line distance in the coordinate plane sense, but if we consider the movement from \((-1,-1)\) to \((3,-3)\) as moving 4 units in the \(x\) - direction and 2 units in the \(y\) - direction and then using the Pythagorean theorem for the path):

$$d=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}\approx 4.47$$

Answer:

\(\sqrt{(3 + 1)^{2}+(-3 + 1)^{2}}=\sqrt{16+4}=\sqrt{20}\approx4.47\) units. If we consider the problem as a simple count of units (assuming movement along the grid for a non - coordinate - geometry strict sense, but based on the coordinate system given): The horizontal distance from \(x=-1\) to \(x = 3\) is \(4\) units and the vertical distance from \(y=-1\) to \(y=-3\) is \(2\) units. Using the Pythagorean theorem (as the path forms a right - triangle), the distance \(d=\sqrt{4^{2}+2^{2}}=\sqrt{20}\approx4.47\) units. So the answer is approximately \(4.47\) units.