A cross-section is a disc with radius 
![V = \int_0^7 A(x)\, dx = \int_0^7 \pi(x+1)^2dx = \int_0^7 \pi(x^2 + 2x + 1)dx \\ \\ = \pi \left[ \frac{1}{3}x^3 + x^2 + x\right]_0^7 = \pi\left( \frac{1}{3}(7)^3 + 7^2 + 7\right) = \frac{511}{3}\pi](https://tex.z-dn.net/?f=V%20%3D%20%5Cint_0%5E7%20A%28x%29%5C%2C%20dx%20%3D%20%5Cint_0%5E7%20%5Cpi%28x%2B1%29%5E2dx%20%3D%20%5Cint_0%5E7%20%5Cpi%28x%5E2%20%2B%202x%20%2B%201%29dx%20%20%5C%5C%20%5C%5C%0A%3D%20%5Cpi%20%5Cleft%5B%20%5Cfrac%7B1%7D%7B3%7Dx%5E3%20%2B%20x%5E2%20%2B%20x%5Cright%5D_0%5E7%20%3D%20%5Cpi%5Cleft%28%20%5Cfrac%7B1%7D%7B3%7D%287%29%5E3%20%2B%207%5E2%20%2B%207%5Cright%29%20%3D%20%5Cfrac%7B511%7D%7B3%7D%5Cpi)
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Answer:
The estimated age of the skull is 39118 years.
Step-by-step explanation:
The amount of the substance after t years is given by:
In which A(0) is the initial amount, and r is the decay rate.
The half-life of carbon-14 is about 5600 years.
This means that . We use this to find r, or 1 - r, to replace in the equation. So
So
Only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire.
This is t for which . So
The estimated age of the skull is 39118 years.