Hi this is Swapnila! I am a Student Success Coach and will be explaining some practice problems from the MA005 Calculus course. Today we will review an assigned problem from Unit 1. In the “Homework Assessment” you were asked to solve this problem:
How far up a wall will a 20 foot long ladder reach if the bottom must be at least 4 feet from the bottom of the wall? What will be the slope of the ladder if the bottom is 4 feet from the wall? What angle will the ladder make with the ground?
Approach this problem as follows:
- Draw a figure using the given information, and define variables.
- Use Pythagorean theorem.
- Find the slope of the ladder
- Use tangent function to find the angle that the ladder makes with the ground because slope is nothing but a tangent of the angle.
Step 1: Draw the figure using the given information as follows:
In the above right angle triangle, the ladder is 4 ft from the bottom of the wall.
Therefore, the horizontal distance = BC = x = 4 ft and ladder is
hypotenuse = AC = z = 20 ft. Also, the top of the ladder is y ft from the bottom of the wall.
Step 2: Use Pythagorean theorem to find the height of the wall AB = y:
By using Pythagorean theorem for the above right angle triangle,
Substitute x=4 and z=20; it becomes
Subtract 16 from both sides; it gives
By using square root property on both sides, it becomes
…… since the distance can never be negative.
Therefore, the ladder could reach 19.6 feet up the wall.
Step 3: Find the slope of the ladder using the formula slope =
From figure and step 2,
Therefore, if the bottom of the ladder is 4 feet from the wall, the slope of the ladder is approximately 4.9.
Step 4: Use tangent function to find angle q because slope is nothing but the tangent of the angle.
Apply arctan on both sides; it becomes
………on the left side arctan and tan cancel with each other.
The ladder could reach 19.6 feet up the wall.
Also, if the bottom of the ladder is 4 feet from the wall, then the slope of the ladder is approximately 4.9 and the angle the ladder makes with the ground is 〖78.5〗^(0 ), which is 1.37 radian.
In addition to this, I am also attaching the pdf document of the practice problem for your ready reference.
Question 9.pdf (270.6 KB)
Please let me know if you have any question on this problem, or on this topic generally. I will be here in the forum for the next hour.