Question

PLEASE ANSWER QUESTION 2(1.) The members of the gardening group plan to build a walkway through the garden as formed by the hypotenuse of each of the four triangles in the drawing. That way, the gardeners will be able to access all sections of the garden. Calculate the length of the entire walkway to the nearest hundredth of a yard. answer: 10 yards(2.)Is the value you just wrote for the total length of the walkway a rational or irrational number? Explain.

Answer 1

We need to compute the hypotenuse of 4 right triangles.

The Pythagorean theorem states:

`c^2=a^2+b^2`

where a and b are the legs and c is the hypotenuse of the right triangle.

In one of the triangles, the length of the legs are: 6 and 8 yards. Then the length of the hypotenuse is:

`\begin{gathered} c^2_1=6^2+8^2 \\ c^2_1=36+64 \\ c_1=\sqrt[]{100} \\ c_1=10yd_{} \end{gathered}`

In another triangle, the length of the legs are: 12 and 8 yards. Then the length of the hypotenuse is:

`\begin{gathered} c^2_2=12^2+8^2 \\ c^2_2=144+64 \\ c_2=\sqrt[]{208} \\ c_2=4\sqrt[]{13}\text{ yd} \end{gathered}`

In the triangle whose hypotenuse (c3) is 15 yd and one of its legs is 12 yd, the unknown is one of the legs, b, which can be computed as follows:

`\begin{gathered} 15^2=12^2+b^2 \\ 225=144+b^2 \\ 225-144=b^2 \\ \sqrt[]{81}=b \\ 9=b \end{gathered}`

The last triangle has legs of 9 yd and 6 yd. Its hypotenuse is:

`\begin{gathered} c^2_4=9^2+6^2 \\ c^2_4=81+36 \\ c_4=\sqrt[]{117} \\ c_4=3\sqrt[]{13} \end{gathered}`

Finally, the length of the walkway is:

`\begin{gathered} c_1+c_2+c_3+c_4=10+4\sqrt[]{13}+15+3\sqrt[]{13}= \\ =(10+25)+(4\sqrt[]{13}+3\sqrt[]{13})= \\ =35+7\sqrt[]{13} \end{gathered}`

This value is irrational because it includes and square root