Chapter 9.6, Problem 7MRPS

Solution

To find: The concrete bags required to build patio.

The number of concrete bags required to construct the patio is 43.

Given data:

The dimensions of the patio are triangular sides of length 8 feet, 11 feet, and 15 feet. And thickness 0.5 ft. One bag of cement takes 0.33 cubic foot of concrete.

Method/Formula used:

  

The ‘Law of Cosines’ states for a triangle ABC , Fig. 1:

  $\cos \angle A=\frac{b^2+c^2-a^2}{2bc}$ … (1)

The area of triangle ABC is

  $\Delta =\frac{1}{2}bc\sin A$ … (2)

Calculations:

The construction volume of triangular patio ABC with sides AB or c equals 15 feet, BC or $a$equals 8 feet and AC or b equals 11 feet, is shown in Fig.2.

  

The volume of concrete used to make the patio is

  $V\text{=Area}\text{of}\text{base}\text{triangle}ABC\text{×thickness}\text{}\text{}\text{of}\text{patio}$ … (3)

Substitute 8 for a , 11 for b and 15 for c in result (1),

  $\cos \angle A=\frac{b^2+c^2-a^2}{2bc}$

  $\begin{array}{c}\cos \angle A=\frac{{11}^2+{15}^2-8^2}{2\left(11\right)\left(15\right)}\\ \angle A={\cos }^{-1}\left(\frac{282}{330}\right)\\ =19.95°\end{array}$

Substitute $19.95°$ for $\angle A$ , 8 for b and 15 for c in result (2), area of triangle ABC is

  $\begin{array}{c}\Delta ABC=\frac{1}{2}\left(11\right)\left(15\right)\sin \left(19.95°\right)\\ =28.15\end{array}$

Thus, the area of the triangular base of the patio is 28.15 sq. feet.

Substitute 28.15 as the area of triangle ABC and 0.5 foot as the height of the patio in result (3), the volume V of concrete used to make the patio is,

  $\begin{array}{c}V\text{=}28.15\text{×0}\text{.5}\\ \text{=}14.075\end{array}$

Thus, the volume of concrete used for constructing the patio is $14.075$ cubic feet.

Since, a concrete bag contains 0.33 cubic feet of the concrete, therefore, the number N of concrete bags required to construct the patio is,

  $N=\frac{\text{}\text{Volume}\text{}\text{}\text{}\text{of}\text{concrete(}V\text{)}}{0.33}$

Substitute $14.075$ for V,

  $\begin{array}{c}N=\frac{\text{}14.075}{0.33}\\ =42.65\\ \approx 43\end{array}$

Therefore, 43 concrete bags are required to construct the patio.