Question

the local pizza restaurant offers free delivery if you live within a 6 - mile radius of the restaurant. the restaurant is located at the origin. adas house is 4 miles west and 5 miles north of the restaurant at point $(-4,5)$. does adas house lie on or within the circle representing the area that gets free delivery?
no, the distance from the restaurant to adas house is $\sqrt{41}$ miles, which is greater than the 6 - mile maximum radius.
yes, the distance from the restaurant to adas house is exactly 6 miles.
no, the distance from the restaurant to adas house is 7 miles, which is greater than the 6 - mile maximum radius.
yes, the distance from the restaurant to adas house is $\sqrt{20}$ miles, which is less than the needed 6 - mile radius.

Explanation:

Step1: 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, the restaurant is at \((0,0)\) (origin) and Ada's house is at \((-4,5)\). So, \(x_1 = 0,y_1 = 0,x_2=-4,y_2 = 5\).
Substitute into the formula: \(d=\sqrt{(- 4-0)^2+(5 - 0)^2}\).

Step2: Simplify the expression

First, calculate \((-4 - 0)^2=(-4)^2 = 16\) and \((5 - 0)^2=25\). Then \(d=\sqrt{16 + 25}=\sqrt{41}\approx6.4\).

Answer:

No, the distance from the restaurant to Ada's house is \(\sqrt{41}\) miles, which is greater than the 6 - mile maximum radius.