Two sides of a triangle have constant lengths a and b, and the angle between them is theta. What value of theta will maximize the area of the triangle?
so far i have the formula and the derivative.
A=.5absin(theta)
A'=.5abcos(theta)
then i set the derivative to zero in order to get the critical point, but i don't know how to solve it because if I divide zero by .5, it will just be 0.
0=.5abcos(theta)
can anyone help me get past this step? thanks.
If A' = 0, and a and b are fixed positive numbers, then cos (theta) = 0 and theta = 90 degrees.
The answer seems intuitively correect.
To find the value of theta that maximizes the area of the triangle, you have correctly taken the derivative of the area formula. Now, let's solve the equation you obtained:
0 = 0.5abcos(theta)
Since you want to solve for theta, you need to isolate it on one side of the equation. Here's how you can proceed:
0.5abcos(theta) = 0
First, divide both sides of the equation by 0.5ab:
cos(theta) = 0
Now, recall that the cosine function equals zero at certain angles. In particular, the cosine of an angle is equal to zero when the angle is pi/2 radians (90 degrees) or 3pi/2 radians (270 degrees). These are the critical points we are looking for.
So, you have two critical points: theta = pi/2 and theta = 3pi/2.
However, we need to check if these critical points correspond to a maximum or minimum for the area of the triangle. To do this, you can take the second derivative.
Take the derivative of the derivative:
A''(theta) = -0.5ab*sin(theta)
Now, let's evaluate this second derivative at theta = pi/2 and theta = 3pi/2:
At theta = pi/2, the second derivative becomes:
A''(pi/2) = -0.5ab*sin(pi/2) = -0.5ab*(-1) = 0.5ab
Since the second derivative is positive at this critical point, it indicates a minimum.
Similarly, at theta = 3pi/2, the second derivative becomes:
A''(3pi/2) = -0.5ab*sin(3pi/2) = -0.5ab*(-1) = 0.5ab
Again, the second derivative is positive, indicating a minimum.
Since the area of a triangle cannot be negative, it means that there is no maximum value for the area of the triangle.
Therefore, the area of the triangle can be increased indefinitely as you increase the angle between the two sides.
In summary, there is no specific value of theta that maximizes the area of the triangle in this case.