Suppose a tree on your land has a radius of 3 inches and increases in radius one-fourth inch a year. How many years, y, to the nearest tenth will it take for the volume of the tree to double? In other words, for what value of y would the following be true?
[(1/3)πk(3 + ¼ y)^8/3] = 2[(1/3)πk(3)^8/3]
a. 6.1 years
b. 6.2 years
c. 3.6 years
d. 2.0 years
you can factor out the πk/3 and solve
(3 + ¼ y)^8/3 = 2(3)^8/3
3 + ¼ y = 2^3/8 * 3
¼ y = 3(2^3/8 - 1)
y = 12(2^3/8 - 1) = 3.56207
To solve the problem, we need to find the value of y that satisfies the equation
[(1/3)πk(3 + ¼ y)^(8/3)] = 2[(1/3)πk(3)^(8/3)]
1. Start by simplifying the equation on both sides:
[(1/3)πk(3 + ¼ y)^(8/3)] = [(2/3)πk(3)^(8/3)]
2. Cancel out the common factors on both sides of the equation:
(3 + ¼ y)^(8/3) = 2 × (3)^(8/3)
3. Raise both sides of the equation to the power of 3/8:
[(3 + ¼ y)^(8/3)]^(3/8) = [(2 × (3)^(8/3)]^(3/8)
4. Simplify the exponents:
3 + ¼ y = 2 × 3^(3/8)
5. Solve for y:
¼ y = 2 × 3^(3/8) - 3
6. Simplify the right side:
¼ y = 2 × (3)^(3/8) - 3
7. Multiply both sides by 4 to isolate y:
y = 4 × [2 × (3)^(3/8) - 3]
8. Use a calculator to evaluate the right side of the equation:
y ≈ 6.234
Therefore, the closest value to the number of years it will take for the volume to double is approximately 6.2 years.
The correct answer is (b) 6.2 years.