Finding missing numbers in a chart.

Question

You are going on an expedition to dig for dinosaur fossils. In digging you
come across a small trunk left by another archeologist. In the trunk is a
paper that lists how deep you need to dig to reach dinosaur fossils.

 

Unfortunately, something spilled on part of the paper and it is damaged.
Using your knowledge of mathematics, find the missing numbers in the
chart. Write a generalization or rule to determine any number of feet you
must dig given the number of years.

 

Years ago        Feet to Dig

10,000                14
20,000                22
30,000                30
40,000

If the Jurassic period was 200 million years ago, how deep would you need
to dig at this location to uncover fossils of that era? Explain why your
answer is or is not reasonable.

Answer 1

First, to solve for the 40,000 year mark.

Let's set up an equation for a line using the first two points, then we will verify that the third point is on the line, then solve for how deep to dig for the 40,000 years.

There are many different basic equations of a line. We have two points on this line (10000,14) and (20000,22) - so it makes the most sense to use the point-slope equation. (y - y₁) = m*(x-x₁)

To find the slope think "rise over run" - i.e. change in y values divided by change in x values.

m = (Δy)/(Δx) = (22-14)/(20000-10000) = (8)/(10000) = 0.0008

For x₁ and y_1, we can pick any value. So let's pick the first value.

(y - 14) = 0.0008 * (x - 10000), which can be simplified down to

y = 0.0008x + 6.

So this is our equation of a line. Let's double check this equation with the third point to make sure we are correct.

30 =?= 0.0008*30000 + 6 = 24 + 6 = 30

30 = 30

So yes, this equation does fit our three points.

Now to solve for 40,000 years.

plug 40,000 in for x, and the resulting answer, y, will be the depth you must dig.

y = 0.0008x + 6 = 0.0008*40000 + 6 = 32 + 6 = 38 feet.

 

Now for 200 million years ago...

Once again, use the equation and plug 200 million in for x to determine the depth.

y = 0.0008x + 6 = 0.0008 * (200,000,000) + 6 = 16000 + 6 = 16006 feet.

A depth of 16006 feet is a little more than 30 miles deep into the ground.

That seems very deep to go to find fossils from that era.

Answer 2

Based on the chart, there is a linear relationship between the number of years and the number of feet.

 

Every 10,000 years is another 8 feet.  If you work that pattern back towards the present, it would be that 0 years ago would be at 6 feet.  That might not make total sense, but at least it helps us find a starting place for the relationship.  In other words, you start at 6 feet for 0 years, and then you add 8 feet for every 10,000 years back:

 

Number of Feet = 8 * (number of 10,000 year periods) + 6

 

 

So what about 200 million years ago?  Divide 200 million by 10,000 and you get 20,000 -- so there are 20,000 periods of 10,000 years in 200 million years.

 

According to the equation, that would be 20,000 * 8 = 160,000 feet.  Since there are 5280 feet in a mile, that would be around 30 miles.  Since that is deeper than anyone has ever dug on Earth, that answer is not reasonable.  (The deepest point that has been reached is 40,230 feet.)