In $2000$, the median home price in New York City was about $\$ 300,000$, and from $2000$ to $2006$, home prices in New York City rose at an average rate of about $11 \%$ per year. If prices had continued to rise at that rate, what would the median home price have been in $2017$? Compare to the actual median price of about $\$ 400,000$ in $2017$.

Assume that poaching reduces the population of an endangered animal by $6 \%$ per year. Further suppose that when the population of this animal falls below $50$ , its extinction is inevitable (owing to the lack of reproductive options without severe in-breeding). If the current population of the animal is $1500$, when will it face extinction? Comment on the validity of the exponential model.

The half-life of carbon-14 is about 5700 years.
a. You find a piece of cloth painted with organic dye. By analyzing the dye, you find that only 77% of the carbon-14 originally in the dye remains. When was the cloth painted?
b. A well-preserved piece of wood found at an archaeological site has 6.2% of the carbon-14 it must have had when it was living. Estimate when the wood was cut.
c. Is carbon14 useful for establishing Earth’s age? Why or why not?

A toxic radioactive substance with a density of $3$ milligrams per square centimeter is detected in the ventilating ducts of a nuclear processing building that was used $55$ years ago. If the half-life of the substance is $20$ years, what was the density of the substance when it was deposited $55$ years ago?

Assume that for the average individual, aspirin has a half-life of $8$ hours in the bloodstream. At $12:00$ noon, you take a $300$-milligram dose of aspirin.

a. How much aspirin will be in your blood at $6:00$ p.m. the same day? at midnight? at $12:00$ noon the next day?

b. Estimate when the amount of aspirin will decay to $5 \%$ of its original amount.

Answer the following question about monthly and annual inflation rates. If the price of gold decreases at a monthly rate of $1.5 \%$, by what percentage does it decrease in a year?

The following situation can be modeled by linear functions. In the following case, draw a graph of the function and use the graph to answer the given question. Be sure you clearly identify the independent and dependent variables. Then briefly discuss whether a linear model is reasonable for the situation described. The diameter of a tree increases by $0.2$ inch with each passing year. When you started observing the tree, its diameter was $4$ inches. Estimate the time at which the tree started growing.

The half-life of C-14 is about 5730 yr.

Archaeologists find a piece of cloth painted with organic dyes. Analysis of the dye in the cloth shows that only 77% of the C-14 originally in the dye remains. When was the cloth painted?

Use the skills covered in the Brief Review on the said page to solve the following equations for the unknown quantity $x$.

$$3^x=99$$

What is logistic growth? Why would it be good if human population growth followed a logistic growth pattern in the future?

Give the slope and y-intercept of each line whose equation is given. Then graph the linear function.

$$f ( x ) = - 3 x + 2$$

Use the radiometric dating formula to answer the following questions.

Uranium-$238$ has a half-life of $4.5$ billion years.

a. You find a rock containing a mixture of uranium-$238$ and lead. You determine that $65 \%$ of the original uranium-$238$ remains; the other $35 \%$ decayed into lead. How old is the rock?

b. Analysis of another rock shows that it contains $45 \%$ of its original uranium-$238$; the other $55 \%$ decayed into lead. How old is the rock?

Use the skills covered in the Brief Review on the said page to solve the following equations for the unknown quantity $x$.

$$10^x=23$$

Use the relative frequency method to answer the following questions.

You toss a coin $100$ times and get only $12$ heads. Do you have reason to suspect the coin is unfair? Explain.