I am not sure how to do this question I know I have to take the derivative but then I am left w/ to unknowns.
If a projectile is shot up hill the distance (range) of the landing point up the hill depends on the initial speed of the projectile,v, the angle of inclination, a, and the angle of elevation of the gun, B. The range of the projectile in metres is given by the formula
R= 2v^2/g [sin(B-a)cosB/cos^2a]
where g is the gravitational field strength, im m/s^2.
(a) Determine the value of B that maximizes the range for a hill with angle of inclination pi over 6(in radians) (30 in degrees)
To find the value of B that maximizes the range, we need to maximize the function R. In this case, R is a function of B.
To do this, we need to take the derivative of the range formula with respect to B and set it equal to zero. Then, we can solve for B.
Let's start by finding the derivative of R with respect to B. To make the calculation easier, we can rewrite the range formula as follows:
R = 2v^2/g * (sin(B - a) * cosB) / cos^2a
Now, let's find the derivative of R with respect to B:
dR/dB = 2v^2/g * [(cos(B - a) * cosB) / cos^2a - sin(B - a) * sinB / cos^2a]
Next, set the derivative equal to zero to find the critical point(s):
dR/dB = 0
2v^2/g * [(cos(B - a) * cosB) / cos^2a - sin(B - a) * sinB / cos^2a] = 0
Now, we can simplify this equation:
[(cos(B - a) * cosB) - sin(B - a) * sinB] = 0
Using the trigonometric identity cos(A - B) = cos(A) * cos(B) + sin(A) * sin(B), we can rewrite the equation:
cos(a) * cosB = sin(a) * sinB
Now, we can solve for B. Divide both sides of the equation by sin(a):
cot(B) = tan(a)
Take the inverse cotangent of both sides:
B = arccot(tan(a))
For our specific case where the angle of inclination is pi/6 radians (30 degrees), substitute a = pi/6 into the equation:
B = arccot(tan(pi/6))
At this point, you can use a calculator or mathematical software to find the exact value of B.
Keep in mind that the derivative test also needs to be done to determine whether the critical point found corresponds to a maximum, minimum, or neither.