Question

When the Moon

is between the Sun and Earth,
it casts a conical shadow called
the umbra. If the shadow is
2140 mi in diameter and 260,955
mi along the edge, what is the
lateral surface area of the umbra?

Answer 1

To find the lateral surface area of the umbra, we need to find the slant height of the cone first. We can use the Pythagorean theorem to do this:

r = 2140/2 = 1070 miles (radius of the base of the cone)
h = 260,955 miles (length along the edge of the cone)

Slant height, s = sqrt(h^2 + r^2) = sqrt((260955)^2 + (1070)^2) = 260,958.7 miles

Now we can use the formula for the lateral surface area of a cone:

Lateral surface area = πrs, where r is the radius of the base and s is the slant height.

Lateral surface area = π(1070)(260958.7) = 883,235,556.5 square miles (rounded to one decimal place)

Therefore, the lateral surface area of the umbra is approximately 883,235,556.5 square miles.