# The rate of change of the angle sum S of a polygon with n sides is a constant 180. If S is 360 when n=4, find S when n=7.

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We are given that `(dS)/(dn)=180` and a point `(S,n)=(360,4)` .

`dS=180dn`

`intdS=int180dn`

`S=180n+C` Using (360,4) to solve for C we get:

`360=180(4)+C==>C=-360`

Thus `S=180n-360`

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**When n=7 we have S=180(7)-360=900**

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Without calculus: If a rate of change is constant the function is linear. The function will be of the form S=180n+k where k is the S intercept. Plugging in known values we get 360=180(4)+k so S=180n-360 as before.

The rate of change is 180 per one change in n.

For n=7 we have n=7-4=3 changes in n.

Therefore angle sum will change 180*3 than n=4 when n=7.

**So when n=7; S= 360+180*3 = 900**

As the rate of change for angle is constant (180) for the angle sum S, therefore the angle sum follows a linear equation that can be written as:

S = 180.n +C where C is a constant and A is the number of angles

for n = 4, S = 360 (given) and substituting this value in the equation we get:

360 = 180*4 + C

=> C = 360-720 = -360

The angle sum is thus given by

S = 180.n-360

for n = 7

S = 180*7-360 = 900

**The angle sum is 900 when n = 7**