a person 6 ft. tall s standing near a street light so that he is 4/10 of the distance from the pole to the tip of his shadow. How high above the ground is the light bulb? if the person's head is exactly 5ft from the light bulb. how far is the person from the pole and how long is the shadow?
as usual, draw a diagram.
If the light is at height h, and the person is standing at a distance of 4x from the pole, then
h/10x = 6/6x
h/10 = 6/6
h=10
So, the light is 10 feet up.
Now, if the person's head is 5' from the light,
(4x)^2 + (10-6)^2 = 5^2
(4x)^2 + 4^2 = 5^2
We all know of the 3-4-5 right triangle, so 4x=3
So, 6x = 4.5, and that is the length of the shadow.
To solve this problem, let's assign some variables:
Let "x" be the distance from the person to the pole
Let "h" be the height of the light bulb
Let "s" be the length of the person's shadow
Given that:
- The person is 6 ft tall.
- The person is standing at a distance 4/10 of the way from the pole to the tip of their shadow.
- The person's head is exactly 5 ft from the light bulb.
Now we can set up the equations:
1) From the given information, we know that the person's shadow is equal to their height. So, we can write:
s = 6 ft
2) We also know that the person is standing at a distance of 4/10 from the pole to the tip of their shadow. Therefore, the person's distance from the pole is (4/10) * s:
x = (4/10) * s
3) We are also given that the person's head is exactly 5 ft from the light bulb, so the person's total height is the same as the distance from the light bulb to the top of their head. This can be represented as:
x + h = 5 ft
Now, let's solve the equations step-by-step:
First, let's substitute the value of "s" from equation 1 into equation 2:
x = (4/10) * 6 ft
x = 2.4 ft
Next, substitute the value of "x" into equation 3:
2.4 ft + h = 5 ft
Now, solve for "h":
h = 5 ft - 2.4 ft
h = 2.6 ft
So, the light bulb is 2.6 ft above the ground.
To find the person's distance from the pole, we already have the value of "x":
x = 2.4 ft
To find the length of the person's shadow, we already have the value of "s":
s = 6 ft
Therefore, the person is 2.4 ft from the pole, and the length of their shadow is 6 ft.
To solve this problem, we can use the concept of similar triangles. Let's label the required lengths:
Height of the person = 6 ft
Distance from the person to the light bulb = x ft
Distance from the person to the pole = y ft
Length of the person's shadow = s ft
Now, let's proceed to solve the problem step by step:
Step 1: Determine the height of the light bulb:
Since the person's head is 5 ft from the light bulb, and the person is 6 ft tall, the light bulb's height above the ground is 6 ft - 5 ft = 1 ft.
Step 2: Find the distance from the person to the pole:
We are given that the person is standing at 4/10 of the distance from the pole to the tip of his shadow. Let's assume the total distance from the pole to the tip of his shadow is d ft. So, the person's distance from the pole is (4/10) * d ft.
Step 3: Find the length of the person's shadow:
Using similar triangles, we can set up the following proportion: (Length of the person's shadow) / (Distance from the person to the pole) = (Height of the person) / (Distance from the person to the light bulb).
We can substitute the given values:
s / ((4/10) * d) = 6 / x
Cross multiplying and simplifying, we get:
10sx = 4d * 6
10sx = 24d
Step 4: Find the value of x:
We know that the person's head is exactly 5 ft from the light bulb. Therefore, the distance from the person to the light bulb is x + 5 ft. Substituting this into the equation above:
10s(x + 5) = 24d
Step 5: Solve for d and s:
We need another equation to solve for the values of d and s individually. Let's consider the similar triangles involving the person, his shadow, and the pole:
Distance from the person to the light bulb = Distance from the pole to the light bulb + Distance from the person to the pole
(x + 5) ft = y + ((4/10) * d) ft
Step 6: Substitute the values obtained in Step 4 and Step 5 into the equation:
(x + 5) ft = y + ((4/10) * d) ft
We know that x + 5 = 5, so the equation becomes:
5 ft = y + ((4/10) * d) ft
Step 7: Simplify the equation:
5 = y + (2/5) * d
Step 8: Solve the system of equations formed in Step 4 and Step 7:
Now, we have two equations:
10s(x + 5) = 24d and 5 = y + (2/5) * d
By solving these two equations simultaneously, we can find the values of x, y, d, and s, which will give us the distance from the person to the pole, the length of the shadow, and the distance from the person to the light bulb.