Question

Jacob is planting shrubs for a landscaping client. He planted more shrubs in the additional region so that the density of shrubs in the new region is equal to the density of shrubs in the rectangular region as shown.

Answer 1

52 shrubs

Explanation:

We know the number of trees planted in a given area, so to calculate the number of total trees, we must calculate the area of the entire figure and thus calculate by means of a proportion

We calculate the total area if we divide the figure in two, as follows:

The area of zone A, which is a rectangle, is calculated as follows:

`\begin{gathered} A_A=b\cdot h \\ A_A=50\cdot20 \\ A_A=1000 \end{gathered}`

The area of zone B, which is a triangle, is calculated as follows:

`A_B=(b\cdot h)/(2)`

To calculate the value of the height of the triangle, we apply the Pythagorean theorem on the right triangle that is formed:

Where the hypotenuse is 25 meters and the other leg is 15 meters, just like this:

`\begin{gathered} h^2=a^2+b^2 \\ 25^2=15^2+b^2 \\ b=\sqrt[]{25^2-15^2} \\ b=\sqrt[]{400} \\ b=20 \\ \text{therefore the height is 20 meters, replacing:} \\ A_B=(30\cdot20)/(2) \\ A_B=300 \end{gathered}`

Now, the total area would be the sum of both areas:

`\begin{gathered} A_T=1000+300 \\ A_T=1300 \end{gathered}`

Which means that if in an area of 400 square meters (20 * 20) they can plant 16 shrubs, in an area of 1300 square meters they would be:

`\begin{gathered} (16)/(400)=(x)/(1300) \\ \text{ solving for x} \\ x=(16\cdot1300)/(400) \\ x=52 \end{gathered}`

In other words, a total of 52 shrubs can be raised throughout the region.