The burning candle

A candle is placed a distance l1 from a thin block of wood of height H. The block is a distance l2 from a wall as shown in Figure 7.1. The candle burns down so that the height of the flame, h1 decreases at the rate of 3 cm/hr. Find the rate at which the length of the shadow y cast by the block on the wall increases. (Note: your answer will be in terms of the constants l1 and l2.

Help please, and thanks!

The rate at which the length of the shadow y increases is given by the formula:

dy/dt = (l1 - l2)/H * dh1/dt

Therefore, the rate at which the length of the shadow y increases is (l1 - l2)/H * 3 cm/hr.

To find the rate at which the length of the shadow cast by the block on the wall increases, we need to use related rates. In this case, we need to find the rate at which the length of the shadow y changes with respect to time.

Let's set up the problem and define the variables given:

l1: distance from the candle to the block
H: height of the block
l2: distance from the block to the wall
h1: height of the flame, which is changing at a rate of -3 cm/hr (negative because it is decreasing)

We are looking for dy/dt, the rate at which the length of the shadow on the wall changes.

From the given information, we can form the following equation:

(l2 + y) / l1 = (H - h1) / h1

Let's rearrange the equation to solve for y:

y = (l1 / h1) * (H - h1) - l2

Now, we can differentiate both sides of the equation with respect to time (t):

dy/dt = (l1 / h1) * [(dH/dt) - (dh1/dt)] - (dl2/dt)

Since dH/dt, dh1/dt, and dl2/dt are not given, we need to express them in terms of the variables given.

dH/dt: The height H of the block is not changing, so dH/dt = 0.

dh1/dt: We are given that h1 decreases at a rate of -3 cm/hr, so dh1/dt = -3 cm/hr.

dl2/dt: The distance l2 from the block to the wall is not changing, so dl2/dt = 0.

Now, let's substitute these values back into the equation:

dy/dt = (l1 / h1) * [(0) - (-3)] - (0)
dy/dt = 3(l1 / h1)

Therefore, the rate at which the length of the shadow y cast by the block on the wall increases is 3(l1 / h1), where l1 and h1 are the constants given.

To solve this problem, we can use similar triangles and the concept of rates.

Let's denote the length of the shadow cast by the block on the wall as y, and the height of the flame as h1.

By similar triangles, we know that the ratio of the length of the shadow to the height of the flame is equal to the ratio of the distance from the block to the wall (l2) to the distance from the candle to the block (l1).

Mathematically, we can write:

y / h1 = l2 / l1

Now, we are asked to find the rate at which the length of the shadow increases. This can be written as dy/dt, where dt represents the change in time.

We are given that the height of the flame decreases at the rate of 3 cm/hr, which can be written as dh1/dt = -3 cm/hr.

To find the rate at which the length of the shadow increases, we can differentiate the equation y/h1 = l2/l1 with respect to time t.

Differentiating both sides of the equation, we get:

(dy/dt * h1 - y * dh1/dt) / h1^2 = 0

Simplifying this equation, we get:

dy/dt = (y * dh1/dt) / h1

Substituting the given values:

dy/dt = (y * (-3)) / h1

Since y/h1 = l2/l1:

dy/dt = (y * (-3)) / (l2/l1)

Simplifying further, we get:

dy/dt = -3 * (y * l1) / l2

Therefore, the rate at which the length of the shadow y cast by the block on the wall increases is given by:

dy/dt = -3 * (y * l1) / l2

Note: The units of y, l1, and l2 must be consistent in order for the equation to be valid.