A fence 3 feet tall runs parallel to a tall building at a distance of 5 feet fro

Question

A fence 3 feet tall runs parallel to a tall building at a distance of 5 feet from the building. We want to find the the length of the shortest ladder that will reach from the ground over the fence to the wall of the building. Here are some hints for finding a solution: Use the angle that the ladder makes with the ground to define the position of the ladder and draw a picture of the ladder leaning against the wall of the building and just touching the top of the fence If the ladder makes an angle 0.33 radians with the ground, touches the top of the fence and just reaches the wall, calculate the distance along the ladder from the ground to the top of the fence____? The distance along the ladder from the top of the fence to the wall is____? Using these hints write a function L(x) which gives the total length of a ladder which touches the ground at an angle x, touches the top of the fence and just reaches the wall. L(x) = ______?. Use this function to find the length of the shortest ladder which will clear the fence. The length of the shortest ladder is_____ feet.

Explanation / Answer

for 0.33 radians distance of ladder till top of fence be l sin 0.33 = 3/l so l = 9.26 m the distance along the ladder from the ground to the top of the fence = 9.26m distance on ground from fence to ladder = 3/ tan0.33 = 8.76m so distance of ladder to wall on ground = 8.76+5 -= 13.76 m so total length of ladder = 13.76/ cos 0.33 = 14.55m so The distance along the ladder from the top of the fence to the wall is = 14.55 - 9.26 = 5.29m let the angle be x length of ladder till fence = 3/sinx length on ground from ladder to fence = 3/tanx length on ground from wall to ladder = 5 + 3/tanx so total length of ladder L(x) = (5tanx+3)/(tanxcosx) = (5tanx+3)/sinx 7.21144/ for shortest ladder differentiate and equate to 0 so [(5sec^2x)(sinx) - cosx(5tanx+3)]/sin^2x = 0 so 5tanx/cosx = 5sinx + 3cosx 5tanx = 5sinxcosx + 3cos^2 x 5sinx(sin^2x/cosx) = 3cos^2 x tan^3 x = 3/5 so x = 40.15 degrees = 0.7 radians so length of shortest ladder = 11.19m

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.