you want to estimate the radius of the towns circular water tank. you stand at point c, about 6 feet from the circular tank. the distance from you to a point of tangency on the tank is about 10 feet. estimate the radius of the tank

Did you mean spherical water tank?

or
water tank is a cylinder with a circular base?

"about 6 feet from the circular tank" ---- 6 feet to the closest point of the tank?
or

" the distance from you to a point of tangency on the tank is about 10 feet"
- if it is a sherical water tank, then there are two tangents, one horizontal and one which is slanted.

????

To estimate the radius of the tank, we can use the properties of a tangent line and a right triangle formed by the distance from point C to the tank and the distance from point C to the point of tangency.

Here's how we can proceed:

Step 1: Draw a diagram to visualize the situation. Sketch a circle to represent the tank and label it with the center O. Place point C outside the circle and connect it to a point of tangency P. Label the distance from C to the tank as 6 feet and the distance from C to P as 10 feet.

C
/
6 / 10
/ .
/ .
. P
.
O

Step 2: Note that the line segment OC connects the center of the circle to the point of tangency P on the circumference. This line segment is perpendicular to the tangent line CP.

Step 3: Recognize that the line segment connecting the center O to the point of tangency P on the circumference is the radius of the circle. Let's call this length r (radius).

Step 4: Use the Pythagorean theorem to find the value of r.

According to the Pythagorean theorem, in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In our case, OC is the hypotenuse, and the other two sides are 6 feet (OC) and 10 feet (CP). So, we have:

OC^2 = CP^2 + OP^2

r^2 = 6^2 + 10^2

r^2 = 36 + 100

r^2 = 136

Taking the square root of both sides, we find:

r ≈ sqrt(136) ≈ 11.66 (rounded to two decimal places)

Therefore, based on the given information, we can estimate the radius of the circular water tank to be approximately 11.66 feet.