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Home Continuous Functions How to Set Up a Problem Examples Example 1: Treasure Map (One Variable)
 
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Example 1: Treasure Map (One Variable)

According to a treasure map, a buried treasure is located due east of a cave and is 200 paces from a tree. The tree is 30 paces east and 40 paces north of the cave. How far is the treasure from the cave?

The solution of this problem uses the quadratic formula, which will be needed throughout the calculus course. We review it here.

We solve Example 1 in three steps.

03_continuous_functions-2.gif

Figure 3.1.1

Step 1

Draw a diagram and label all quantities involved. In Figure 3.1.1, we put the cave at the origin and let x be the distance from the cave to the target along the x-axis. The tree is at the point (30, 40), and the treasure is at the point (x, 0).

Step 2

Write the known information as a system of formulas. By the distance formula, we have

200 =03_continuous_functions-3.gif, x ≥ 0.

The inequality x ≥ 0 arises because the treasure is east of the cave.

Step 3

Solve for x, We square the Distance Formula.

40,000 = (x - 30)2 + (0 - 40)2 = x2 - 60x + 900 + 1600 =

= x2 - 60x + 2500 x2 - 60x - 37,500 = 0

To find x we use the Quadratic Formula.

03_continuous_functions-4.gif

INTERPRET THE SOLUTION

Since x ≥ 0, we reject the negative solution. Thus

x = 30 + 03_continuous_functions-5.gif ~ 226 paces.

The treasure is approximately 226 paces from the cave.

Most calculus problems involve two or more variables.
Home Continuous Functions How to Set Up a Problem Examples Example 1: Treasure Map (One Variable)

Last Update: 2006-11-15