A woman 2.00 m tall stands 3.5 m in front of a plane mirror. (a) What is the minimum height the mirror must be to allow the woman to view her complete image from head to foot? Assume that her eyes are 10 cm below the top of her head.(b) What would be the required minimum height of the mirror if she were to stand 5.2 m away?
To find the minimum height of the mirror required for the woman to view her complete image, we can use the concept of similar triangles.
a) In this case, we have a right-angle triangle formed by the woman's height, her distance from the mirror, and the distance from her eyes to the top of her head. Let's denote the minimum height of the mirror as "h" and set up the following proportion:
woman's height / distance from mirror = (woman's height - distance from eyes to top of head) / distance from mirror + height of mirror
We can plug in the given values:
woman's height = 2.00 m
distance from mirror = 3.5 m
distance from eyes to top of head = 0.1 m (10 cm)
Now, let's solve the proportion for "h":
2.00 m / 3.5 m = (2.00 m - 0.1 m) / (3.5 m + h)
Cross-multiply the equation to get:
(2.00 m) * (3.5 m + h) = (2.00 m - 0.1 m) * (3.5 m)
Simplify:
7.00 m^2 + 2.00 m * h = 6.72 m^2
Rearrange the equation to find "h":
2.00 m * h = 6.72 m^2 - 7.00 m^2
2.00 m * h = -0.28 m^2
h = -0.28 m^2 / 2 m
h = -0.14 m (Since we can't have a negative height, we ignore this result)
Since we can't have a negative height, the minimum height of the mirror required for the woman to view her complete image from head to foot is not possible in this scenario.
b) Now let's consider the case when the woman stands 5.2 m away from the mirror. We can follow the same approach as before and set up the proportion:
woman's height / distance from mirror = (woman's height - distance from eyes to top of head) / distance from mirror + height of mirror
Plugging in the given values:
woman's height = 2.00 m
distance from mirror = 5.2 m
distance from eyes to top of head = 0.1 m (10 cm)
Let's solve the proportion for "h":
2.00 m / 5.2 m = (2.00 m - 0.1 m) / (5.2 m + h)
Cross-multiply the equation to get:
(2.00 m) * (5.2 m + h) = (2.00 m - 0.1 m) * (5.2 m)
Simplify:
10.4 m^2 + 2.00 m * h = 10 m^2
Rearrange the equation to find "h":
2.00 m * h = 10 m^2 - 10.4 m^2
2.00 m * h = -0.4 m^2
h = -0.4 m^2 / 2 m
h = -0.2 m (Since we can't have a negative height, we ignore this result)
Similarly, for this scenario, the minimum height of the mirror required for the woman to view her complete image from head to foot is not possible.
In both cases, it is not possible to find a mirror height that allows the woman to see her complete image from head to foot.