A town sits at point (5,1) on a map. From the tallest building in the town, you can see objects up to 7 miles away. If each unit on the map is equivalent to 1 mile, you can see a statue in a town at point (2,5)? Explain.

The distance between the tallest building and the statue is 5 miles, so as the statue is inside the range of 7 miles, it can be seen from the tallest building.

Step-by-step explanation:

To solve the problem we need to know the distance between the tallest building in the town and the statue.

This distance can be calculated with the formula of the distance between two points:

d = sqrt(dx^2 + dy^2), where dx is the difference in x-coordinate between the points and dy is the difference in y-coordinate between the points.

So, the distance between these points is:

dx = 5-2 = 3

dy = 1-5 = -4

d = sqrt(3^2 + (-4)^2) = sqrt(9+16) = 5 miles

The distance is 5 miles, so as it is inside the range of 7 miles, the statue can be seen from this point.

e^(-3x) and e^(4x) form a fundamental set of solutions of the differential equation

y'' - y' - 12y = 0.

Since their wronskian is 7e^x and not 0

Step-by-step explanation:

Given the differential equation

y'' - y' - 12y = 0.

We are required to verify that the functions e^(-3x) and e^(4x) form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]). They form a fundamental set if they are linearly independent, and they are linearly independent if their wronkian is not zero, otherwise, they are linearly dependent.

Now, we need to find the Wronskian of e^(-3x) and e^(4x), and see if it is equal to zero or not.

The wrinkles of two functions y1 and y2 is given as the determinant

W(y1, y2) = |y1y2|

|y1'y2'|

W(e^(-3x), e^(4x))

= |e^(-3x)e^(4x)|

...|-3e^(-3x)4e^(4x)|

= e^(-3x) × 4e^(4x) - (-3e^(-3x) × e^(4x))

= 4e^x + 3e^x

= 7e^x

≠ 0

Since the wronskian is not zero, we conclude that the are linearly independent, and hence, form a set of fundamental solution.