A minor makes claim to a circular piece of land with a radius of 40 m from a given point, and is entitled to dig to a depth of 25 m. If the minor can dig tunnels at any angle, find the length of the longest straight tunnel that he can dig, to the nearest metre.
draw a side view.
The diameter is 80 and the height is 25, so the diagonal is
√(80^2 + 25^2)
FYI:
a minor is a person under legal age
a miner is one who mines minerals
Haha yeah just a silly spelling mistake. sorry
Well, well, well! Looks like the minor is a real underground adventurer, huh? Alright, let's calculate the length of the longest straight tunnel they can dig.
Now, imagine the circular piece of land as a yummy, giant pizza. The minor is going to dig their way through this pizza, but they can only go as deep as 25 m. So, let's cut ourselves a slice and take a closer look.
If we draw a vertical line from the center of the circle to the edge, we get two right-angled triangles. One triangle with a height of 25 m and a hypotenuse of 40 m.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the base of this triangle, which will be the longest straight tunnel the minor can dig.
So, applying the glorious theorem, we have:
base^2 + 25^2 = 40^2
base^2 + 625 = 1600
base^2 = 1600 - 625
base^2 = 975
Now, let's bust out our comedic calculator real quick to find the length of the base... *drum roll*... It's approximately 31 meters!
So, my friend, the minor can dig a straight tunnel that's about 31 meters long. Just be careful not to get lost down there!
To find the length of the longest straight tunnel that the minor can dig, we can use the Pythagorean theorem. The length of the tunnel will be the hypotenuse of a right triangle, with the radius of the circular piece of land as one leg, and the depth of the tunnel as the other leg.
Using the Pythagorean theorem, the length of the tunnel (h) can be found using the formula:
h = √(r^2 + d^2)
Where:
h = length of the tunnel
r = radius of the circular piece of land (40 m)
d = depth of the tunnel (25 m)
Plugging in the values:
h = √(40^2 + 25^2)
h = √(1600 + 625)
h = √(2225)
h ≈ 47.17
Therefore, the length of the longest straight tunnel that the minor can dig is approximately 47 meters.
To find the length of the longest straight tunnel that the minor can dig, we can use a geometric approach.
First, let's visualize the situation. We have a circular piece of land with a radius of 40 m. The minor is entitled to dig a tunnel with a depth of 25 m. We want to find the length of the longest straight tunnel that can be dug within this circular area.
Now, let's break down the problem into smaller parts. Imagine a right triangle within the circular area, with the radius as the hypotenuse and the depth as one of the legs. The other leg of the triangle represents the length of the longest straight tunnel that can be dug.
Using the Pythagorean theorem, we can find the length of this leg. The formula is:
c^2 = a^2 + b^2
where c is the hypotenuse (radius of the circle), a is the depth of the tunnel, and b is the length of the tunnel.
Rearranging the formula, we have:
b^2 = c^2 - a^2
Substituting the values we know:
b^2 = (40 m)^2 - (25 m)^2
b^2 = 1600 m^2 - 625 m^2
b^2 = 975 m^2
Taking the square root of both sides, we find:
b ≈ √975 m
Calculating this, we get:
b ≈ 31.23 m
Therefore, the length of the longest straight tunnel that the minor can dig, to the nearest meter, is approximately 31 meters.