A sign 3 feet high is placed on a wall with its base 2 feet above the eye level of a woman attempting
to read it. How far from the wall should she stand to get the best view of the sign, that is, so that the
angle subtended at the woman’s eye by the sign is a maximum?
Hint: there is a right triangle with one side of 3 feet + 2 feet. If the woman moves away from the sign, distance of the bottom of the right triangle changes. Angle of the side of 3 feet + 2 feet changes as well.
correction: angle opposite of the side of 3 feet + 2 feet changes as well.
To find the distance from the wall the woman should stand to get the best view of the sign, we need to maximize the angle subtended at her eye by the sign.
Let's set up a diagram to help us visualize the problem. Let "x" represent the distance from the wall the woman should stand, and let "h" represent the height of the sign (3 feet). Since the sign's base is 2 feet above the eye level of the woman, the height of the sign relative to her eye level is (3 - 2) = 1 foot.
A
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h | \
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x
The angle subtended at the woman's eye by the sign can be determined using trigonometry. Since we have a right triangle formed by the height of the sign, the distance from the wall, and the line connecting the woman's eye to the top of the sign, we can use the tangent function.
tan(theta) = opposite/adjacent
= h / x
Now, to maximize the angle, we need to find the value of "x" that maximizes the ratio h / x.
To do this, we can take the derivative of h / x with respect to x and set it equal to zero.
d(h / x) / dx = 0
Differentiating h / x with respect to x, we get:
(dh / dx) / x - h / x² = 0
Since dh / dx is 0 (as the height of the sign doesn't change), we can simplify the equation to:
- h / x² = 0
Rearranging the equation:
h = 0
However, this is not possible as it contradicts our problem statement. Hence, there is no maximum point. Instead, the angle subtended at the woman's eye by the sign will keep increasing as she moves further away from the wall. Therefore, the woman should stand as far back as possible for the best view of the sign.