A convex lens of a focal length f can be defined by the lens equation 1/f=1/p + 1/q if an object is a distance p from the lens, then the distance q from the lens to the images. For a particular lens, f=2cm and p is increasing find
1. a general formula for the rate of change of q w.r.t. p.
2. the rate of change of q w.r.t. p=22cm
a. 1/q=1/f-1/p
1/g= (p+f)/pf
q= pf/(p+f)
dq/dp= f/(p+f)-pf/(p+f)^2 (remember f is a constant, so df/dp=0)
check that.
is part 2 1/144
To find the rate of change of q with respect to p, we can differentiate the lens equation 1/f = 1/p + 1/q with respect to p.
1. Differentiate the lens equation with respect to p:
d/dp (1/f) = d/dp (1/p) + d/dp (1/q)
Since f is a constant, the derivative of 1/f with respect to p is zero.
0 = -1/p^2 + d/dp (1/q)
2. Rewrite the equation to isolate the derivative of q with respect to p:
d/dp (1/q) = 1/p^2
The rate of change of q with respect to p is given by the equation:
d(q)/d(p) = 1/p^2
Note: The rate of change of q with respect to p does not depend on the specific value of f. It only depends on the position of the object, which is determined by p.
Now, to find the specific rate of change when p = 22 cm:
1/p^2 = 1/(22)^2 = 1/484
Therefore, the rate of change of q with respect to p when p = 22 cm is 1/484 cm.
To find the rate of change of q with respect to p, we can simply take the derivative of the lens equation with respect to p. Let's break down the steps:
1. Start with the lens equation:
1/f = 1/p + 1/q
2. Multiply both sides of the equation by pq to eliminate the fractions:
pq/f = q + p
3. Rearrange the equation to solve for q:
q = (pf)/(p-f)
4. Now, let's differentiate both sides of the equation with respect to p:
d(q)/d(p) = d((pf)/(p-f))/d(p)
5. To differentiate the right-hand side, we can use the quotient rule:
d(q)/d(p) = [(p)(d(f)/(d(p))) - (f)(d(p)/(d(p)))] / (p-f)^2
6. Simplify the equation:
d(q)/d(p) = (pd(f)/d(p) - f) / (p - f)^2
Now, let's calculate the rate of change of q with respect to p at the specific value p = 22 cm:
1. Substitute the given value p = 22 cm into the equation:
d(q)/d(p) = (22d(f)/d(p) - 2) / (22 - 2)^2
2. Since the focal length f is given as f = 2 cm, we can substitute it into the equation:
d(q)/d(p) = (22d(f)/d(p) - 2) / (20)^2
3. Finally, substitute the given rate of change d(q)/d(p) = 22 cm into the equation:
22 = (22d(f)/d(p) - 2) / 400
4. Solve for d(f)/d(p):
22 × 400 = 22d(f)/d(p) - 2
8800 + 2 = 22d(f)/d(p)
8802 = 22d(f)/d(p)
d(f)/d(p) = 8802/22
d(f)/d(p) = 400
Therefore, the rate of change of q with respect to p when d(q)/d(p) = 22 cm is equal to 400.