14) A spot Light is on the ground $$ (20 \mathrm{ft} $$ away from a wall and 6 FT ) person is walking towards the wall at a rate of $$ 2.5 \mathrm{tT} / \mathrm{sec} $$. How fast is the height of the shadow changing when the person is 8 fT from the wall? Is the shadow increasing or decreasing in height at this Time?

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UpStudy AI · Accepted Answer

To solve this problem, we can use related rates in calculus. Let's define the variables involved:

- Let $$ h $$ be the height of the shadow.
- Let $$ x $$ be the distance of the person from the wall.
- The distance from the spotlight to the wall is a constant $$ 20 $$ ft.
- The height of the person is $$ 6 $$ ft.
- The person is walking towards the wall at a rate of $$ \frac{dx}{dt} = -2.5 $$ ft/sec (negative because $$ x $$ is decreasing).

### Step 1: Set up the relationship

Using similar triangles, we can set up the following relationship:

$$
\frac{h}{20} = \frac{6}{x}
$$

From this, we can express $$ h $$ in terms of $$ x $$:

$$
h = \frac{6}{20} x = \frac{3}{10} x
$$

### Step 2: Differentiate with respect to time

Now, we differentiate both sides with respect to time $$ t $$:

$$
\frac{dh}{dt} = \frac{3}{10} \frac{dx}{dt}
$$

### Step 3: Substitute known values

We need to find $$ \frac{dh}{dt} $$ when the person is $$ 8 $$ ft from the wall. At that moment, $$ x = 8 $$ ft. We also know $$ \frac{dx}{dt} = -2.5 $$ ft/sec.

Now we can substitute $$ \frac{dx}{dt} $$ into the differentiated equation:

$$
\frac{dh}{dt} = \frac{3}{10} (-2.5)
$$

### Step 4: Calculate $$ \frac{dh}{dt} $$

Now, let's calculate $$ \frac{dh}{dt} $$:

$$
\frac{dh}{dt} = \frac{3}{10} \times -2.5 = -\frac{7.5}{10} = -0.75 \text{ ft/sec}
$$

### Conclusion

The height of the shadow is changing at a rate of $$ -0.75 $$ ft/sec when the person is $$ 8 $$ ft from the wall. Since the rate is negative, the shadow is decreasing in height at this time.
When the person is 8 ft from the wall, the height of the shadow is decreasing at a rate of 0.75 ft/sec.

A spot Light is on the ground away from a wall and 6 FT )
person is walking towards the wall at a rate of . How fast is
the height of the shadow changing when the person is 8 fT from the wall?
Is the shadow increasing or decreasing in height at this Time?

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A spot Light is on the ground away from a wall and 6 FT ) person is walking towards the wall at a rate of . How fast is the height of the shadow changing when the person is 8 fT from the wall? Is the shadow increasing or decreasing in height at this Time?