Question

Mr. Jackson has just finished building his treehouse and still needs to buy a ladder to be attached to the ledge of the treehouse and anchored at a point on the ground, as modeled below. Mr. Jackson is standing 1.3 meters from the stilt supporting the treehouse. This is the point on the ground where he has decided to anchor the ladder. The angle of elevation from his eye level to the bottom of the treehouse is 56 degrees. Mr. Jackson's eye level is 1.5 meters above the ground. Determine and state the minimum length of a ladder, to the nearest tenth of a meter, that Mr. Jackson will need to buy for his treehouse. (do not round until the very end)

Answer 1

The length of the ladder Mr. Jackson will need to buy for his treehouse is approximately 3.7 meters

Explanation:

The parameters given in the question are;

The horizontal distance from the point Mr. Jackson has decided to anchor the ladder to the the stilt supporting the tree house = 1.3 meters

The angle of elevation from Mr. Jackson's eye level to the bottom of the treehouse = 56°

The height of Mr. Jackson's eye level from the ground = 1.5 meters

From the question, we can draw the geometry of the ladder attached to the treehouse using Microsoft Visio as shown in the attached diagram

From the diagram, by trigonometric ratios, we have;

tan(56°) = a/1.3

∴ a = 1.3 × tan(56°) ≈ 1.927

By Pythagoras' theorem, we have;

The length of the ladder = √((a + b)² + 1.3²)

From the drawing, b = 1.5 m

Therefore, we have

The length of the ladder = √((1.927 + 1.5)² + 1.3²) ≈ 3.6653

The length of the ladder without rounding = √((1.3 × tan(56°) + 1.5)² + 1.3²) = 3.66559488352

∴ The length of the ladder given to the nearest tenth of a meter ≈ 3.7 m

The length of the ladder Mr. Jackson will need to buy for his treehouse ≈ 3.7 m.