To determine the location and height of the image formed by a convex mirror, we can use the mirror equation:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
where:
f = focal length of the mirror (radius of curvature)
d_o = object distance from the mirror
d_i = image distance from the mirror
In this case, we are given:
f = 1.60 x 10^2 cm
d_o = -40.0 cm (negative because the object is in front of the mirror)
To find the image distance (d_i), we need to rearrange the mirror equation:
\[ \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \]
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} - \frac{1}{-40.0} \]
Calculating this expression will give us the image distance.
Once we know the image distance, we can use the magnification formula to find the height of the image:
\[ \text{magnification (m)} = - \frac{d_i}{d_o} \]
Substituting the known values into the formula, we can find the height of the image:
\[ \text{height of image} = \text{magnification} \times \text{height of object} \]
Let's calculate the values step by step:
First, let's find the image distance (d_i):
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} - \frac{1}{-40.0} \]
To evaluate this expression, we calculate the reciprocal of each value on the right-hand side of the equation, then add them together, and finally take the reciprocal of that sum.
\[ \frac{1}{d_i} = \frac{1}{1.60\times 10^2} + \frac{1}{40.0} \]
Now, we calculate the sum:
\[ \frac{1}{d_i} = 0.00625 + 0.025 \]
\[ \frac{1}{d_i} = 0.03125 \]
Taking the reciprocal of both sides gives:
\[ d_i = \frac{1}{0.03125} \]
This evaluates to:
\[ d_i = 32 \text{ cm} \]
So the image distance is 32.0 cm.
Now, let's find the magnification:
\[ \text{m} = - \frac{d_i}{d_o} = - \frac{32}{-40.0} \]
This gives us:
\[ \text{m} = 0.8 \]
Finally, we can find the height of the image:
\[ \text{height of image} = \text{magnification} \times \text{height of object} = 0.8 \times 11.0 \]
Thus, the height of the image is 8.8 cm.
To summarize:
(a) The location of the image is 32.0 cm from the convex mirror.
(b) The height of the image is 8.8 cm.