A child holds a candy bar 13.4 cm in front of the convex side-view mirror of an automobile.
The image height is reduced by one fifth.
What is the radius of curvature of the mirror?
Answer in units of cm.
I did 1/5=-image distance (i)/13.4-->i=-2.68
1/f=1/-2.68+1/13.4---->-3.35
|f|*2=r
r=6.7cm
but this was not the right answer. Where did I go wrong/ how do you solve?
To solve this problem, we'll start by using the mirror formula:
1/f = 1/u + 1/v
where:
f is the focal length of the mirror,
u is the object distance from the mirror, and
v is the image distance from the mirror.
Given that the image height is reduced by one-fifth, it means the image distance (v) is one-fifth of the object distance (u).
So, if the object distance (u) is 13.4 cm, the image distance (v) would be (1/5)*13.4 cm = 2.68 cm.
Substituting these values into the mirror formula:
1/f = 1/13.4 + 1/2.68
Simplifying this equation gives us:
1/f = 0.0746 + 0.3731
1/f = 0.4477
Now, to find the radius of curvature (r):
r = 2*f
Substituting the value of f into the equation:
r = 2 * (1/0.4477)
r = 2.2375
Therefore, the radius of curvature of the mirror is approximately 2.2375 cm.
To solve this problem, you need to use the mirror formula for convex mirrors:
1/f = 1/v + 1/u,
where f is the focal length of the mirror, v is the image distance, and u is the object distance.
In this case, the image height is reduced by one fifth, which means the image distance (v) is 4/5 times the object distance (u):
v = (4/5)u.
Now, let's substitute this value of v into the mirror formula:
1/f = 1/(4/5)u + 1/u.
To solve for f, we need to find the common denominator and simplify the equation:
1/f = (5/4u + 1/u) / (4/5).
Now, invert the denominator and multiply across:
1/f = (5u + 4u) / (4u) * (5/4).
Simplifying further:
1/f = 9u / 4u * (5/4).
Now, cancel out the common factors:
1/f = 9 / 4 * (5/4).
Multiply the numerators and denominators:
1/f = 45/16.
To find f, take the reciprocal of both sides:
f = 16/45.
The radius of curvature (r) of the mirror is twice the focal length (r = 2f):
r = 2 * (16/45).
Multiply across:
r = 32/45.
Therefore, the radius of curvature of the mirror is 32/45 cm.