Question

30 POINTS!!! HELP PLEASE!

Model each scenario with an equation and a sketch. Solve for the missing value and u. Use complete sentences to interpret the solution. In your final answer, include your equation, sketch, interpretation, and all calculations necessary for the solution.

1.) To get home from school, Bob walks four blocks north and three blocks east. What is the straight line distance between Bob’s house and his school?

2.)One of the requirements of your summer window-washing job is to provide yourself with all of the necessary supplies, including a fourteen foot ladder. When you arrive at your first job, you place your ladder on the ground six feet from the base of the house and lean it towards a second story window, only to realize that the ladder doesn’t reach the window. Given the length of the ladder and its current position, what is one possible height of the second story window? (Hint: There is more than one correct answer.)

3.) In a softball diamond, each of the bases, including home plate, are equidistant from each other. Although the name implies differently, a softball diamond is in the shape of a square. Given that the distance between the bases is unknown, determine an expression for the straight line distance between first and third bases.

4.) A sailboat drifts 600 meters west, makes a turn and sails 800 meters south. How far is the sailboat from its original position?

Answer 1

#1) In the sketch, the school is located at coordinates (1, 1). Going north (up) 4 blocks and east (right) 3 blocks to get home places it at (4, 5). The distance formula is:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`. Using our coordinates we have:

`d=√((4-1)^2+(5-1)^2) \\=√(3^2+4^2) \\=√(9+16) \\=√(25) \\=5` His house is 5 blocks from the school in a straight line.
#2) The ladder (14 ft) forms the hypotenuse of a right triangle, with the legs being the distance from the house (6 ft) and the height of the ladder (b ft). This gives us:

`6^2+b^2=14^2 \\36+b^2=196`
Subtract 36 from both sides:

`36+b^2-36=196-36 \\b^2=160`
Take the square root of both sides:

`√(b^2)=√(160) \\b=12.6`
One possibility for the height of the second story window is 13 feet, since it is longer than 12.6.
#3) The distance from 1st base to 3rd base forms the hypotenuse of a right triangle, with each leg being equal to s, the side length of the square formed by the baseball diamond. Using the Pythagorean theorem we have:

`s^2+s^2=d^2`, where d is the distance from 1st to 3rd. Combining like terms gives us
`2s^2=d^2`. Taking the square root of both sides we have
`√(2s^2)=√(d^2) \\s√(2)=d` (this is due to the fact that square root cancels a squared variable, so s comes out of the radical sign).
#4) The straight distance from the boat's original position to its new position forms the hypotenuse of a right triangle, with the legs being the distance west (600) and the distance south (800) it traveled. Using the Pythagorean theorem we have:

`600^2+800^2=d^2 \\360000+640000=d^2 \\1000000=d^2`
Taking the square root of both sides we have

`√(1000000)=√(d^2) \\1000=d`
The straight distance between the two points would be 1000 meters.