Calculating the Distance between the Feet of a Stepladder: Sin Rule & Pythagoras

In summary, the two sides of a stepladder are of length 2m and 1.85m, with the longer side inclined at 65° to the horizontal when fully opened. Using the sin rule and Pythagoras' theorem, the distance apart of the feet of the step ladder can be found to be 3.45m. However, it is important to note that this calculation assumes a 90° angle at the top of the ladder when fully opened, which may not be accurate.
  • #1
MrPickle
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Homework Statement


The two sides of a stepladder are of length 2m and 1.85m. When fully opened, the longer side is inclined at 65° to the horizontal. Find the distance apart of the feet of the step ladder.

Homework Equations


Sin Rule & Pythagoras?

The Attempt at a Solution


I had a go but I really think it's wrong.

HomeworkQues.jpg


[itex]\frac{2}{\sin90°} = \frac{b}{\sin65°}[/itex]

[itex]b = \frac{2}{\sin90°}\times \sin65°[/itex]

[itex]b = 1.81 (2d.p.)[/itex]

[itex]a^2 = 2^2 - 1.81^2 = 0.7239[/itex]

[itex]a = 0.85 (2d.p.)[/itex]

[itex]x^2 = 1.85^2 - 0.85^2 = 2.7[/itex]

[itex]x = 1.64 (2d.p.)[/itex]

[itex]l = 1.81 + 1.64 = 3.45m[/itex]

P.S. How do you do new lines in LaTeX?
 
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  • #2


How do you know that the top angle is 90 degrees when it is fully opened? I think you have to stick with just the 65 degree angle and the leg lengths in order to find the distance.

Also, I don't know how to do new lines in LateX. I always just close the tex tag and start a new one.
 
  • #3


I thought it would be 90 degrees because it says 65 degree to the horizontal.
 
  • #4


You can't assume that the ladder opens up at 90 degrees (it doesn't actually).
 
  • #5


I see what I did wrong. I got horizontal and verticle mixed up so I drew my diagram wrong.
 

What is the "Sin Rule" in mathematics?

The Sin Rule, also known as the Law of Sines, is a mathematical rule used to solve triangles with known angle measures and side lengths. It states that the ratio of a side length to the sine of its opposite angle is equal for all sides and angles in a triangle. This rule can be written as a/sinA = b/sinB = c/sinC, where a, b, and c are the side lengths and A, B, and C are the corresponding angles.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental mathematical principle that states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be written as c^2 = a^2 + b^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

How do you use the Sin Rule to solve a triangle?

To use the Sin Rule to solve a triangle, you must have at least one known angle and its opposite side length. You can then set up and solve a proportion using the ratios of side lengths to their corresponding sine values. If you have two known angles and one side length, you can use the fact that the sum of the angles in a triangle is 180 degrees to find the third angle, and then solve the proportion as before. If you have two known side lengths and an angle opposite one of them, you can use the Pythagorean Theorem to find the third side length, and then solve the proportion as before.

What is the relationship between the Sin Rule and the Pythagorean Theorem?

The Sin Rule and the Pythagorean Theorem are both rules used to solve triangles. The Sin Rule is used for solving triangles with known angle measures and side lengths, while the Pythagorean Theorem is used for solving right triangles. However, the Pythagorean Theorem can also be used in conjunction with the Sin Rule to solve non-right triangles, as mentioned in the previous question.

Why are the Sin Rule and the Pythagorean Theorem important in geometry and trigonometry?

The Sin Rule and the Pythagorean Theorem are fundamental principles in geometry and trigonometry that are used to solve a wide variety of problems involving triangles. They are essential tools in calculating distances, heights, and angles in real-world scenarios such as construction, navigation, and engineering. They also form the basis for more advanced mathematical concepts and are crucial for understanding and applying trigonometric functions and identities.

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