A light is placed 5 ft. from the center of a globe 3 ft. in diameter. Find the area of the illuminated portion.
thank you!
Why did the light have to be placed so far from the center of the globe? It's like the globe didn't want to be in the spotlight! Anyway, let's shed some light on the situation!
To find the area of the illuminated portion, we need to calculate the area of the circular disc formed by the shadow of the globe on the ground.
Since the globe has a diameter of 3 ft, its radius would be 1.5 ft. Now, if we draw a radius from the center of the globe to the edge of the circular shadow, we'll form a right triangle.
The hypotenuse of this triangle would be the distance from the globe's center to the light source, which is 5 ft. And since the radius of the globe is the adjacent side, we can use trigonometry to find the opposite side.
Using the Pythagorean theorem, we can find the length of the opposite side:
opposite side = sqrt(hypotenuse^2 - adjacent side^2)
= sqrt(5^2 - 1.5^2)
= sqrt(25 - 2.25)
= sqrt(22.75)
≈ 4.768 ft
Now that we have the length of the opposite side, we can find the angle formed between the light source and the edge of the shadow. This angle can be calculated using the inverse tangent function:
angle = arctan(opposite side / adjacent side)
= arctan(4.768 / 1.5)
≈ 71.57 degrees
So, the illuminated portion forms a sector of the circular shadow with this angle. To find the area of this sector, we can use the formula:
area of sector = (angle / 360) * π * radius^2
= (71.57 / 360) * π * 1.5^2
≈ 0.389 π ft^2
Therefore, the area of the illuminated portion is approximately 0.389 π square feet.
To find the area of the illuminated portion, we need to determine the area of the portion of the globe that is within the illuminated region.
Step 1: Calculate the distance from the light source to the outer edge of the illuminated region.
The distance from the light source to the outer edge of the illuminated region is the sum of the radius of the globe and the distance from the center of the globe to the light source.
Radius of the globe = 1.5 ft (diameter divided by 2)
Distance from the center of the globe to the light source = 5 ft.
Distance from the light source to the outer edge of the illuminated region = Radius of the globe + Distance from the center of the globe to the light source = 1.5 ft + 5 ft = 6.5 ft.
Step 2: Calculate the area of the entire globe.
The area of a sphere is given by the formula: A = 4πr², where r is the radius.
Radius of the globe = 1.5 ft
Area of the globe = 4π(1.5 ft)² = 4π(2.25 ft²) = 9π ft².
Step 3: Calculate the area of the portion of the globe that is within the illuminated region.
The area of the portion of the globe that is within the illuminated region is proportional to the square of the ratio of the distance from the light source to the outer edge of the illuminated region to the radius of the globe.
Ratio = (Distance from the light source to the outer edge of the illuminated region) / (Radius of the globe) = 6.5 ft / 1.5 ft = 4.333
Area of the illuminated portion = (Ratio)² × Area of the globe = 4.333² × 9π ft² = 80.721π ft².
Therefore, the area of the illuminated portion is approximately 80.721π square feet (or 254.578 ft² if you prefer a decimal approximation).
Please note that the answer is rounded to three decimal places.
To find the area of the illuminated portion, we can first find the area of the entire globe and then subtract the area of the unilluminated portion.
1. Find the area of the entire globe:
The formula for the surface area of a sphere is A = 4πr^2, where A is the surface area and r is the radius. The radius of the globe is half the diameter, so r = 3ft / 2 = 1.5ft.
Plugging in the value of the radius, we get A = 4π(1.5)^2.
2. Find the area of the unilluminated portion:
We want to find the area of the part of the sphere that is not illuminated by the light. This is the area of the hemisphere that is opposite to the light source. The radius of this hemisphere is equal to the distance from the light source to the center of the globe, which is 5ft.
The formula for the surface area of a hemisphere is A = 2πr^2, where A is the surface area and r is the radius. Plugging in the value of the radius, we get A = 2π(5)^2.
3. Calculate the area of the illuminated portion:
The area of the illuminated portion is the difference between the area of the entire globe and the area of the unilluminated portion.
Area of illuminated portion = Area of globe - Area of unilluminated portion.
Area of illuminated portion = 4π(1.5)^2 - 2π(5)^2.
Simplifying this expression will give you the area of the illuminated portion.
make a side view of the event
draw a line from the light to the centre of the sphere,
draw a line to the sphere representing your line of sight, this would be a tangent to the sphere
It is easy to see, that this tangent would be 4 ft
(the 3-4-5 right-angled triangle)
the angle Ø at the centre of that triangle is sin^-1 (4/5) = appr 53.13°
so we need the surface area of the "cap".
let x be the distance from centre to base of the cap
cos 53.13 = x/3
.6 = x/3
x = 1.8
so the height , h, of the cap is 3-1.8 = 1.2
The curved surface area of the spherical cap is 2πrh
where r is the radius of the sphere and h is the height of the cap
SA = 2π(3)(1.2) = 7.2π square feet