I am having a bit of trouble getting started with this problem:
Any help would be appreciated.The hypotenuse of a right-angled triangle is 12 cm in length. Find the measures of the angles in the triangle that maximize its perimeter.
I am having a bit of trouble getting started with this problem:
Any help would be appreciated.The hypotenuse of a right-angled triangle is 12 cm in length. Find the measures of the angles in the triangle that maximize its perimeter.
Right Angle Triangle
Perimeter = Side1 + S2 + S3
Side 1 = 12cm (hypotenuse)
S2 = 12cosX (if you draw a right angled triangle you can figure this out through basic trig)
S3 = 12sinX
Therefore;
P(x) = Side1 + S2 + S3
P(x) = 12 + 12CosX + 12SinX
To find max; use calculus!
P'(x) = 12(-sinX) + 12(cosX)
P'(x) = 12(cosX - sinX)
Max when this = 0
P'(x) = 0
12(cosX - sinX) = 0
therefore; cosX - sinX = 0
therefore; cosX = sinX
Only place these are = to eachother is at 45 degrees, or Pi/4 radians.
To check this is a max. Upon graphing 0 < x < 90 (angle cannot be less than or zero, or cannot be more than or equal to 90 degrees)
Graph
0_______ (positive)_________45 (0)_______________(negative)_____________90
transitions from + to - therefore Perimeter is maximised at 45 degrees.
Hope this helped