A street light is at the top of a 19 foot tall pole. A 6 foot tall woman walks away from the pole with a speed of 7 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 feet from the base of the pole?
Let her distance from the pole be x ft
let the length of her shadow by y ft
given: dx/dt = 7 ft/s
find: d(x+y)/dt when x = 30
by ratios:
19/(x+y) =6/y
19y = 6x+6y
13y = 6x
13dy/dt = 6dx/dt
dy/dt = 6(7)/13 = 42/13 ft/s
(notice dy/dt is a constant , which means the shadow is lengthening and moving at a rate independent of where she is)
d(x+y)/dt = 7 + 42/13 = 133/13 or appr 10.23 ft/s
Well, well, well! Looks like we have a shadow chasing conundrum here! Let's shed some light on this, shall we?
We have a triangle formed by the street light, the pole, and the woman's shadow. The height of the pole is 19 feet, and the woman is 6 feet tall. So, the length of the shadow is the hypotenuse of this triangle.
Taking a stroll with a speed of 7 ft/sec, our woman is moving away from the pole. We need to find out how fast the tip of her shadow is moving when she's 30 feet from the base.
To solve this, we can use a little math magic called similar triangles. The triangles formed by the woman, the pole, and the shadow are similar, meaning their angles are the same. This means the ratio of the lengths of their corresponding sides is always the same.
So, the ratio of the length of the shadow to the distance from the base is constant. Let's call this ratio "x".
We have two similar triangles: one with sides of 19 (pole height), 6 (woman's height), and x (length of the shadow), and the other with sides of 30 (distance from the base), 6 (woman's height), and y (rate of change of the shadow's length).
Using proportions, we can set up the equation: 19/6 = x/y
Simplifying, we get 19y = 6x
Now, let's differentiate both sides of the equation with respect to time (t), since we want to know the rate of change:
19(dy/dt) = 6(dx/dt)
We're given that the woman's rate of change of distance is 7 ft/sec, so dx/dt = 7.
Plugging that in, we have 19(dy/dt) = 6(7).
Simplifying further, dy/dt = (6 * 7) / 19.
So, the rate at which the tip of her shadow is moving when she is 30 feet from the base of the pole is approximately 2.63 ft/sec.
Now, that wasn't shady at all, was it?
To find the rate at which the tip of the woman's shadow is moving, we need to use similar triangles.
Let's denote:
- Height of the pole as h (19 feet)
- Height of the woman as w (6 feet)
- Distance between the woman and the pole as x (30 feet)
We can create two similar triangles: one formed by the pole, the shadow, and the ground, and another formed by the woman, her shadow, and the ground.
Using the similar triangles, we can set up the following proportion:
(woman's height) / (distance from woman to pole) = (length of woman's shadow) / (length of the pole's shadow)
In mathematical terms, this can be expressed as:
w / x = (h + w) / s
To find s (length of the pole's shadow), we can rearrange the equation as:
s = (w * x) / (h + w)
Now, we can differentiate this equation with respect to time to find the rate at which the pole's shadow is changing when the woman is 30 feet from the base of the pole.
Differentiating both sides of the equation:
ds/dt = (d/dt)(w * x) / (h + w)
Given that the woman is walking away from the pole at a speed of 7 ft/sec (dw/dt = 7 ft/sec) and we need to find the rate at which the pole's shadow is changing (ds/dt), we can substitute these values into the equation:
ds/dt = (d/dt)(w * x) / (h + w)
ds/dt = (w * dx/dt + x * dw/dt) / (h + w)
Substituting the known values into the equation:
ds/dt = (6 * 7 + 30 * 0) / (19 + 6)
ds/dt = 42 / 25
Therefore, the rate at which the tip of the woman's shadow is moving when she is 30 feet from the base of the pole is 42/25 ft/sec.
To find the speed at which the tip of her shadow is moving, we need to use similar triangles and related rates. Let's break down the problem step by step.
1. First, let's draw a diagram to visualize the situation. We have a pole with a street light at the top, a woman walking away from the pole, and her shadow on the ground.
2. Notice that the shadow forms a right triangle with the woman's path and the pole. Let's label the length of the shadow as 's', the height of the pole as 'h', and the distance from the woman to the base of the pole as 'x'.
3. From the given information, we know that the height of the pole is 19 feet and the woman is 6 feet tall.
4. As the woman walks away from the pole, the shadow grows longer. We need to find how fast the shadow is changing when she is 30 feet from the base of the pole.
5. To relate the lengths of the sides of the triangles, we can use similar triangles. The small triangle formed by the woman, her shadow, and the ground is similar to the large triangle formed by the pole, its shadow, and the ground.
6. The ratio of the corresponding sides of similar triangles is equal. So we have: s / (s + x) = h / (h + 6).
7. Now, let's differentiate both sides of the equation with respect to time since we want to find the rate of change. We'll let ds/dt represent the rate at which the shadow is changing and dx/dt represent the rate at which the woman is moving. Note that dh/dt is 0 since the height of the pole remains constant.
8. Differentiating the equation gives us: (ds/dt * (s + x) - s * dx/dt) / (s + x)^2 = 0 / (h + 6)^2.
9. Since dh/dt is 0, the equation simplifies to: (ds/dt * (s + x) - s * dx/dt) / (s + x)^2 = 0.
10. Rearranging the equation, we have: ds/dt = (s * dx/dt) / (s + x).
11. Now, we know that the woman's speed, dx/dt, is 7 ft/sec. We need to find the value of s when x is 30 feet.
12. To find s, we can substitute the values into the similar triangles equation we derived earlier: s / (s + x) = h / (h + 6).
13. Plugging in the values, we get: s / (s + 30) = 19 / 25.
14. Cross-multiplying the equation, we have: 25s = 19(s + 30).
15. Simplifying, we get: 25s = 19s + 570.
16. Solving for s, we find: s = 114/6 = 19 feet.
17. Finally, substituting s = 19 feet and dx/dt = 7 ft/sec into the equation we derived in step 10, we get: ds/dt = (19 * 7) / (19 + 30) = 133/49 ≈ 2.71 ft/sec.
Therefore, the tip of the woman's shadow is moving at a speed of approximately 2.71 ft/sec when she is 30 feet from the base of the pole.