A 5 ft long man walks away from the foot of a 12(½) ft high lamp post at the rate of 3 mph. What will be the rate at which the shadow increases?

The correct option is 2mph.

To find the rate at which the shadow increases, we can set up a relationship between the distance the man has walked and the length of his shadow using similar triangles.

Let:
- $x$ be the distance of the man from the foot of the lamp post at any time $t$.
- $y$ be the length of the man's shadow at any time $t$.

We are given the following information:
- Height of the lamp post = $12\frac{1}{2}$ ft = $12.5$ ft.
- Height of the man = $5$ ft.
- The rate at which the man walks away from the lamp post is:
$$\frac{dx}{dt}=3\text{ mph}$$

The lamp post and the man are both vertical, forming two similar right-angled triangles with the ground and the ray of light.
By the property of similar triangles, the ratio of their heights is equal to the ratio of their bases:
$$\frac{\text{Height of lamp post}}{\text{Total distance from lamp post to tip of shadow}}=\frac{\text{Height of man}}{\text{Length of shadow}}$$
Substituting the values:
$$\frac{12.5}{x+y}=\frac{5}{y}$$

Now, we cross-multiply to solve for $y$ in terms of $x$:
$$12.5y=5(x+y)$$
$$12.5y=5x+5y$$
Subtract $5y$ from both sides:
$$7.5y=5x$$
$$y=\frac{5}{7.5}x=\frac{2}{3}x$$

To find the rate at which the shadow is increasing, we differentiate both sides with respect to time $t$:
$$\frac{dy}{dt}=\frac{2}{3}·\frac{dx}{dt}$$

Substitute the given value of $\frac{dx}{dt}=3$ mph:
$$\frac{dy}{dt}=\frac{2}{3}·3=2\text{ mph}$$

Thus, the rate at which the shadow increases is 2 mph.