24 Ladders In One
Distance Formula – Advanced Algebra Help Please?
My daughter has finally passed my level of math comprehension. Could some one please explain how begin the first part of this following problem? This chapter is dealing with the distance formula.
10.
A 24 ft ladder is placed upright against a wall. Then the top of the ladder slides down the wall while the foot of the ladder is pulled outward along the ground at a steady rate of 2 ft/s.
a. Find the heights that the ladder reaches at 1 s intervals while the ladder slides down the wall.
b. How long will it take before the ladder is lying on the ground?
c. Does the top of the ladder also slide down the wall at a steady rate of 2 ft/s? Explain your reasoning.
d. Write parametric equations that model the distance in feet of the foot of the ladder from the wall, x, and the height in feet that the top of the ladder reaches, y. Let t represent time in seconds.
Initially the top of the ladder is 24 feet up the wall. The bottom of the ladder is pulled out at 2 feet/sec so we have a set of right-triangle problems with the hypotenuse always being the 24 ft ladder. x is the distance of the bottom of the ladder from the wall. y is the distance from the bottom of the wall to the point the ladder is leaning on now. x*x + y*y = 24*24 if you remember your Pythagorean theorem as sung by Danny Kay. x starts at 0 and goes up 2 ft each second. y= sqrt(576 – x*x)
question a- second by second:
seconds x y
0 0 24
1 2 23.9
2 4 23.7
3 6 23.2
4 8 22.6
5 10 21.8
6 12 20.8
7 14 19.5
8 16 17.8
9 18 15.9
10 20 13.3
11 22 9.6
12 24 0 Bang! The ladder is now flat on the ground so we can stop calculating.
b. 12 seconds
c. Looking at the numbers for y from part a, it is clear that in the first few seconds the top of the ladder moved down much slower than 2 feet per second, In the first half of the 12 seconds, the top has moved down much less than half way. In the final seconds it was coming down fast, so the average speed was 2 feet/second, but it was *not* a steady 2 feet/second for the 12 seconds.
d. Distance of the foot of the ladder from the wall is x=2*t.
Height of the top of the ladder is y=sqrt((24^2)-((2*t)^2))
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