Buried treasure.
Ahmed has half of a treasure map,which indicates that the treasure is buried in the desert 2
x+ 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk
xpaces to the north, and then walk 2
x+ 4 paces to the east. If they share their information, then they can find
xand save a lot of digging. What is
x?
Requirements: Show each step of your mathwork and explain each step. View attached example very similar to this question on how to complete this assignment.
- Your solution to the above problem, making sure to include all mathematical work, and an explanation for each step.
- If there is more than one solution to the quadratic equation, be sure you indicate which one is the best solution for this application problem and why. Translate the polynomial measurements of the triangle to paces according to your solution.
- Incorporate the following four math vocabulary words into your writing.
Zero factor
- Pythagorean Theorem
- Compound equation
INSTRUCTOR GUIDANCE EXAMPLE: Week 5 Written Assignment INSTRUCTOR GUIDANCE EXAMPLE: Week Five Assignment Pythagorean Quadratic Here is a treasure hunting problem very similar to the one in the textbook. Spanky has half of a treasure map which indicates treasure is buried 2x + 9 paces from Leaning Rock. Buckwheat has the other half of the treasure map which says that to find the treasure one must walk x paces to the north from Leaning Rock, and then 2x + 6 paces east. Spanky and Buckwheat found out that with both bits of information they can solve for x and save themselves a lot of digging. How many paces is x? Even though Spanky’s half of the map doesn’t indicate which direction the 2x + 9 paces should go, we can assume that his and Buckwheat’s paces should end up in the same place. When we sketch this out on scratch paper we see that it forms a right triangle with 2x + 9 being the length of the hypotenuse, and x and 2x + 6 being the legs of the triangle. Now we know how we can use the Pythagorean Theorem to help solve for x. The Pythagorean Theorem states that in every right triangle with legs of length a and b and hypotenuse c, these lengths have the relationship of a 2 + b 2 = c 2 . Let a = x, and b = 2x + 6, so that c = 2x + 9. Then, by putting these measurements into the Theorem equation we have x 2 + (2x + 6) 2 = (2x + 9) 2 The binomials into the Pyth. Thrm. x 2 + 4x 2 + 24x + 36 = 4x 2 + 36x + 81 The binomials squared. Notice there is a 4x 2 on both sides of the equation which can be -4x 2 -4x 2 subtracted out first. x 2 + 24x + 36 = 36x + 81 Subtract 36x from both sides of equation. -36x -36x x 2 – 12x + 36 = 81 Subtract 81 from both sides of equation. -81 -81 x 2 – 12x – 45 = 0 Now we have a quadratic equation to solve by factoring and using the zero factor. (x – ) (x + ) = 0 Since the coefficient of x 2 is 1 we can start with a pair of parenthesis with an x in each. Since the 45 is negative we know there will be one + and one – in the binomials. We need two factors of -45 which add up to -12. -1, 45; -3, 15; -5, 9 1, -45; 3, -15; 5, -9. Looks like 3, -15 will do it! (x – 15)(x + 3) = 0 Use the zero factor property to solve each binomial, x – 15 = 0 or x + 3 = 0 creating a compound equation. x = 15 or x = -3 These are the possible solutions to our equation. However, one of these solutions is what we call extraneous because it doesn’t work with this scenario at all. You cannot have negative paces or negative distance in a measured geometric figure, so the -3 solution does not work, leaving us with x = 15 as the key number of paces. The treasure lies 15 paces north and 2x + 6 = 2(15) + 6 = 36 paces east of Leaning Rock, or 2x + 9 = 2(15) + 9 = 39 paces straight from the rock as the crow flies! [Students will need to provide the usual mechanics of a paper along with appropriate introductory and concluding paragraphs.]