Explanation:

To find the fourth proportional of 2a, a3, and c, we need to use the concept of proportion.

Proportion

A proportion is an equality between two ratios. For example, if a/b = c/d, then we can say that a, b, c, and d are in proportion. We can also write this as a:b::c:d, where the double colon (::) represents proportionality.

Formula for Fourth Proportional

If a, b, and c are in proportion, then the fourth proportional will be:

d = (b * c) / a

Using the Formula

Let's use the formula to find the fourth proportional of 2a, a3, and c:

a : a3 :: 2a : c

We can rewrite this proportion as:

a/a3 = 2a/c

To find the fourth proportional, we need to multiply the second and third terms and divide by the first term:

d = (a3 * 2a) / a

Simplifying this expression, we get:

d = 2a2

Therefore, the fourth proportional to 2a, a3, and c is 2a2.

Conclusion

In conclusion, we used the concept of proportion and the formula for the fourth proportional to find the fourth proportional of 2a, a3, and c. The fourth proportional is 2a2.