Question

The slope of a ladder making an angle ${60^ \circ }$ with floor is ___

A. 1
B. $- \sqrt 3$
C. $- \dfrac{1}{{\sqrt 3 }}$
D. $\sqrt 3$

Answer 1

Hint : As we can see in the diagram the vertical change is the opposite side of the angle ${60^ \circ }$ and horizontal change is the adjacent side to the angle ${60^ \circ }$ . The ratio of vertical change and horizontal change is the slope which is equal to the ratio of opposite side and adjacent side to the angle ${60^ \circ }$ . We already know that the ratio of opposite side and adjacent side in a right angled triangle is tangent to the angle. So find tangent to the angle ${60^ \circ }$ to get the value of slope.

Complete step-by-step answer :
We are given to find the slope of a ladder making an angle ${60^ \circ }$ with the floor.
Slope is a number which describes both the steepness and direction of a line. It is calculated by dividing the vertical change with horizontal change between any two distinct points on a line.

From the diagram, we can say that slope is $\dfrac{{\Delta x}}{{\Delta y}}$
But as we can see the ladder with the floor and normal forms a right angled triangle.
In a right triangle, the ratio of opposite side to the adjacent side of an angle gives a tangent function.
Here $\Delta x$ is the opposite side and $\Delta y$ is the adjacent side to the angle ${60^ \circ }$
Therefore, Slope is $\dfrac{{\Delta x}}{{\Delta y}} = \tan {60^ \circ }$
The value of $\tan {60^ \circ }$ is $\sqrt 3$
Therefore, the slope of the ladder making an angle ${60^ \circ }$ with floor is $\sqrt 3$
So, the correct answer is “Option D”.

Note : Another approach for finding the value of $\tan {60^ \circ }$
Tangent function is the ratio of sine function to the cosine function.
So to find $\tan {60^ \circ }$ we need the values of $\sin {60^ \circ }$ and $\cos {60^ \circ }$
$\sin {60^ \circ } = \dfrac{{\sqrt 3 }}{2},\cos {60^ \circ } = \dfrac{1}{2}$
$\tan {60^ \circ } = \dfrac{{\left( {\dfrac{{\sqrt 3 }}{2}} \right)}}{{\left( {\dfrac{1}{2}} \right)}} = \dfrac{{\sqrt 3 }}{1} = \sqrt 3$