Question

The diagram shows the side view of a shed. Calculate the height of the shed.

Answer 1

Hint: Here, we will first use the Pythagorean theorem \[{h^2} = {a^2} + {b^2}\], where \[h\] is the hypotenuse, \[a\] is the height and \[b\] is the base of the triangle. Apply this formula, and then use the given conditions to find the required value.

Complete step-by-step answer:
Let us assume that the perpendicular height of the roof of a shed is \[x\] m and \[y\] m is the half of the base of the shed.
We will now draw the diagram of the shed with the height of the roof and \[y\] m.

Since we know that the \[y\] m is half of the base of the shed, which is \[3.2\] m.
We will first find the value of \[y\] in the given figure.
\[
   \Rightarrow y = \dfrac{{3.2}}{2} \\
   \Rightarrow y = 1.6 \\
 \]
We will use the Pythagorean theorem \[{h^2} = {a^2} + {b^2}\], where \[h\] is the hypotenuse, \[a\] is the height and \[b\] is the base of the triangle.
Applying the Pythagorean theorem on the roof of the shed, we get
\[
   \Rightarrow {2^2} = {x^2} + {\left( {1.6} \right)^2} \\
   \Rightarrow 4 = {x^2} + 2.56 \\
 \]
Subtracting the above equation by \[2.56\] on each of the sides, we get
\[
   \Rightarrow 4 - 2.56 = {x^2} + 2.56 - 2.56 \\
   \Rightarrow 1.44 = {x^2} \\
   \Rightarrow {x^2} = 1.44 \\
 \]
Taking the square root on both sides in the above equation, we get
\[
   \Rightarrow x = \sqrt {1.44} \\
   \Rightarrow x = \pm 1.2 \\
 \]
Since the height of the roof of the shed can never be negative, the negative value of \[x\] is discarded.
Thus, the height of the roof of the shed is \[1.2\] m.
We know that the total height of the shed is the sum of height of the base of the shed and the height of the roof of the shed.
We will now find the height of the shed from the above values.

\[
  {\text{Height}} = 2.4 + 1.2 \\
   = 3.6 \\
 \]

Therefore, the height of the shed is \[3.6\] m.

Note: In solving these types of questions, one may note that it is necessary to take all the dimensions in the same unit otherwise it will be the wrong approach. The key step for solving this problem is the knowledge of Pythagorean theorem, \[{h^2} = {a^2} + {b^2}\], where \[h\] is the hypotenuse, \[a\] is the height and \[b\] is the base of the triangle, in geometrical mathematics, we can calculate the value of height of this system in seconds.