Aschool, hospital, and a supermarket are located at the vertices of a right triangle formed by three highways. the school and hospital are 14.7 miles apart. the distance between the school and the supermarket is 8.82 miles, and the distance between the hospital and the supermarket is 11.76 miles.

a service road will be constructed from the main entrance of the supermarket to the highway that connects the school and hospital. what is the shortest possible length for the service road? round to the nearest tenth.

7.1 miles

Step-by-step explanation:

Consider right triangle HospitalSchoolSupermarket. In this triangle:

The shortest road from the main entrance of the supermarket to the highway that connects the school and hospital will be the height drawn from the point Supermarket to the hypotenuse HospitalSchool.

Let the length of this road be x mi and the distance from School to point A be y mi. Use twice the Pythagorean theorem for right triangles Supermarket SchoolA and SupermarketHospitalA:

Subtract from the second equation the first one:

Thus,

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: