To get to the library from the bank you drive due north and then east. The distance you drove due east is 14 more miles than the distance you drove north.  The straight line distance between the...

To get to the library from the bank you drive due north and then east. The distance you drove due east is 14 more miles than the distance you drove north.  The straight line distance between the library and the bank is 26 miles.  How far did you drive?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The length of a straight line from the bank to the library is 26 miles. To get to the library the person first drives north and then east. The distance driven due east is 14 miles more than that driven north.

Let the distance driven due north be D, the distance driven due east is D + 14.

D^2 + (D + 14)^2 = 26^2

=> D^2 + D^2 + 196 + 28D = 676

=> 2D^2 + 28D - 480 = 0

=> D^2 + 14D - 240 = 0

=> D^2 + 24D - 10D - 240 = 0

=> D(D + 24) - 10(D + 24) = 0

=> (D - 10)(D + 24) = 0

=> D = 10, D = -24

As the distance traveled cannot be negative take only the positive root D = 10

The total distance traveled is 10 + 10 + 14 = 34 miles.

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steveschoen's profile pic

steveschoen | College Teacher | (Level 1) Associate Educator

Posted on

(D+14)^2 means (D+14)(D+14)

That gives us, after FOIL-ing, D^2 + 14D + 14D + 196 =

                                                    D^2 + 28D + 196

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steamgirl | Student, College Junior | (Level 1) Honors

Posted on

To start this problem, draw a small compass so that you can see that North and East are perpendicular to one another. Then draw a line starting from their intersection point out, an label it being 26 miles. Now draw a dotted line from the end of your new line down to the east, x-axis, line. Then label that the distance north (change in the y direction) as x and the distance east (change in the x direction) as x +14. You now have a right triangle and can use the Pythagorean Theorem. 

x^2 + (x+14)^2 = 26^2

Simplify:

x^2 + (x^2 + 28x +196) = 676

2x^2 +28x -480 = 0

x^2 +14x - 240 = 0

Factor:

(x-10)(x+24)=0

Which leaves you with x = 10, -24. Distance cannot be negative, therefore, -24 is not a valid answer. Plug in 10 to find that the total distance is x +(x+14) = 34 miles.

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Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

The driving was done on the two sides of a right triangle. The distance between the bank and the library is the slanted side, the hypotenuse. If you think about the cardinal directions, you will realize why this must be the case.

Lets say the distance you drove north is "  x " . Then the distance you drove east is " x + 4 " . We need to find a way to relate this to the hypotenuse.

We use the Pythagorean theorem : a^2 + b^2 = c^2

Substitute in our expressions to get :

x^2 + (x + 14)^2 = 26^2

Then solve for x.

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mchandrea | Student, College Junior | (Level 1) Salutatorian

Posted on

Drawing the problem always help me understand. Refer to attached photo. In this case, the bank is at south and the library at east. It can be seen that a triangle is formed. The longest side was also given. I used the pythagorean theorem to solve this. 

Two values for x was solved. The first one was -24. Since distance can never be negative, it is safe to assume that the correct answer is the second value which is 10. 

To get the final answer, the distance that you drove would be x + x + 14. Substituting 10 as the value for x, the final answer is 34 miles.

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nisarg's profile pic

nisarg | Student, Grade 11 | (Level 1) Valedictorian

Posted on

To get to the library from the bank you drive due north and then east. The distance you drove due east is 14 more miles than the distance you drove north.  The straight line distance between the library and the bank is 26 miles.  How far did you drive?

This makes a triangle because you go up "x" mile and straight east "x+14" miles

the hypotenuse is 26 miles so this can be solved with the Pythagorean theorem

`26^2=x^2+(x+14)^2`  

676=`x^2+x^2+28x+196`

`676=2x^2+28x+196`

`338=x^2+14x+98`

`x^2+14x-240=0`

`(x-10)+(x+24)=0`

`x=-24 x=10`

Since distance cannot be negative the amount driven north is 10 miles

The total distance would be 10miles north+24miles east so you drove 34 miles total

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rachellopez | Student, Grade 12 | (Level 1) Valedictorian

Posted on

I think the best way to answer this question is by drawing a picture (visuals always help). When you draw the picture you can see that the lines created form a triangle. To find x you will have to use the Pythagorean Theorem,       a^2 + b^2 = c^2, c being the hypotenuse (your straight side).

(x)^2 + (x+14)^2 = (26)^2                   Plug in your numbers

x^2 + (x^2 + 28x + 196) = 676             Square your terms

2x^2 + 28x - 480 = 0                            Combine like terms to set equal to 0

2 (x + 24) (x - 10)                                  Factor

2 `!=` 0    x = -24   x = 10                         Set factors equal to 0` `

Since 2 obviously cannot equal 0 that option is eliminated. Distances cannot be negative so x will not equal -24. Your final answer is x = 10.This means you traveled 10 miles north and 24 miles (10 + 14) east for a total of 34 miles driven.

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