Problem:
A corn cob is shaped somewhat like a cone, with the radius of its broadest end as 2.1 cm and length as 20 cm. If each 1 cm of the surface of the cob carries an average of four grains, how many grains would you find on the entire cob?

Solution:

Step 1: Finding the Volume of the Corn Cob
To find the number of grains on the entire cob, we first need to calculate the volume of the corn cob.

The volume of a cone can be calculated using the formula:
V = (1/3) * π * r^2 * h

Where:
V = Volume of the cone
π = Pi, approximately 3.14
r = Radius of the broadest end of the cone
h = Height or length of the cone

Given values:
r = 2.1 cm
h = 20 cm

Using the formula, we can calculate the volume of the corn cob:

V = (1/3) * 3.14 * (2.1 cm)^2 * 20 cm
V ≈ 3.14 * 4.41 cm^2 * 20 cm
V ≈ 277.2 cm^3

Therefore, the volume of the corn cob is approximately 277.2 cm^3.

Step 2: Finding the Surface Area of the Corn Cob
Next, we need to calculate the surface area of the corn cob. Since each 1 cm of the surface carries an average of four grains, we can determine the number of grains on the entire surface.

The surface area of a cone can be calculated using the formula:
A = π * r * (r + l)

Where:
A = Surface area of the cone
π = Pi, approximately 3.14
r = Radius of the broadest end of the cone
l = Slant height of the cone

To find the slant height, we can use the Pythagorean theorem:
l = √(r^2 + h^2)

Given values:
r = 2.1 cm
h = 20 cm

Calculating the slant height:
l = √((2.1 cm)^2 + (20 cm)^2)
l ≈ √(4.41 cm^2 + 400 cm^2)
l ≈ √(404.41 cm^2)
l ≈ 20.11 cm

Using the surface area formula, we can calculate the surface area of the corn cob:

A = 3.14 * 2.1 cm * (2.1 cm + 20.11 cm)
A ≈ 3.14 * 2.1 cm * 22.21 cm
A ≈ 146.67 cm^2

Therefore, the surface area of the corn cob is approximately 146.67 cm^2.

Step 3: Calculating the Number of Grains
Now that we have the surface area of the corn cob, we can determine the number of grains on the entire cob.

Since each 1 cm of the surface carries an average of four grains, the total number of grains can be calculated by multiplying the surface area by four:

Number of grains = 146.67 cm^2 * 4
Number of grains