Q44. - Algebra 2

Question

Mr. Cardona is a regional sales manager for a company in Florida. Tampa, Orlando, and Ocala outline his region. If a coordinate grid in which 1 unit = 10 miles is placed over a map of Florida with Tampa at the origin, the coordinates of the three cities are (0,0), (7,5), (2,5, 10). Use a determinant to estimate the area of his sales territory.

Step-by-Step Solution

Answer

Area of the sales territory is 287.5 square miles.

Step 1 - Define area of triangle

The area of triangle having vertices at $\left(a,b\right)$, $\left(c,d\right)$ and $\left(e,f\right)$ is $\left|A\right|$, where $A=\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ c& d& 1\\ e& f& 1\end{array}\right|$.

Here vertices of the triangle are $\left(a,b\right)=\left(0,0\right)$, $\left(c,d\right)=\left(7,5\right)$ and $\left(e,f\right)=\left(2.5,10\right)$. So, substitute 0 for a, 0 for b, 7 for c, 5 for d, 2.5 for e and 10 for f into the expression $A=\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ c& d& 1\\ e& f& 1\end{array}\right|$.

$A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 7& 5& 1\\ 2.5& 10& 1\end{array}\right|$

Step 2 - Find the determinant

Find the determinant in this case by using expansion by minors.

 

$\begin{array}{c}A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 7& 5& 1\\ 2.5& 10& 1\end{array}\right|\\ =\frac{1}{2}\left(\left(0\right)\left|\begin{array}{cc}5& 1\\ 10& 1\end{array}\right|-\left(0\right)\left|\begin{array}{cc}7& 1\\ 2.5& 1\end{array}\right|+\left|\begin{array}{cc}7& 5\\ 2.5& 10\end{array}\right|\right)\end{array}$

Step 3 - Evaluate 2 × 2 determinants

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is.,  $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$. 

Apply this definition to find the $2\times 2$ determinant.

 

$\begin{array}{c}A=\frac{1}{2}\left(0-0+\left[7\left(10\right)-5\left(2.5\right)\right]\right)\\ =\frac{1}{2}\left(70-\text{12.5}\right)\\ =\frac{1}{2}\left(\text{57.5}\right)\\ =\text{28.75}\end{array}$

 

Since the area of the triangle is an absolute value, therefore, are of the triangle is 28.75 square units.

Step 4 - Estimate area of sales territory

Since the area of the triangle is formed by three vertices is 28.75 square units and it is given that 1 unit = 10 miles, therefore, the area of sales territory will be

 

$\Rightarrow 28.75\times 10=287.5$square miles.

 

Therefore, area is 287.5 square miles.