Question

a path goes around a triangular park, as shown.
39
a. find the distance around the park to the nearest yard.
the distance is about 50 yards.
b. a new path and a bridge are constructed from point q to the mid - point m of (overline{pr}). find qm to the nearest yard.
qm≈□yd
c. a man jogs from p to q to m to r to q and back to p at an average speed of 150 yards per minute. to the nearest tenth of a minute, about how long does it take him to travel the entire distance?
it takes about □ minutes.

Explanation:

Step1: Find coordinates of points

Assume \(P=(0,50)\), \(Q=(0,0)\), \(R=(80,0)\).

Step2: Calculate length of \(PQ\)

Using distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for \(P=(0,50)\) and \(Q=(0,0)\), \(PQ=\sqrt{(0 - 0)^2+(0 - 50)^2}=50\).

Step3: Calculate length of \(QR\)

For \(Q=(0,0)\) and \(R=(80,0)\), \(QR=\sqrt{(80 - 0)^2+(0 - 0)^2}=80\).

Step4: Calculate length of \(PR\)

For \(P=(0,50)\) and \(R=(80,0)\), \(PR=\sqrt{(80 - 0)^2+(0 - 50)^2}=\sqrt{6400 + 2500}=\sqrt{8900}\approx94.3\).

Step5: Find perimeter for part a

Perimeter \(=PQ + QR+PR=50 + 80+94.3 = 224.3\approx224\) yards.

Step6: Find mid - point \(M\) of \(PR\)

Mid - point formula \(M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\), for \(P=(0,50)\) and \(R=(80,0)\), \(M=(\frac{0 + 80}{2},\frac{50+0}{2})=(40,25)\).

Step7: Calculate \(QM\) for part b

For \(Q=(0,0)\) and \(M=(40,25)\), \(QM=\sqrt{(40 - 0)^2+(25 - 0)^2}=\sqrt{1600 + 625}=\sqrt{2225}\approx47.2\approx47\) yards.

Step8: Calculate total distance for part c

Total distance \(d = PQ+QM + MR+RQ+QP=50+47+47+80+50 = 274\) yards.

Step9: Calculate time

Using \(t=\frac{d}{v}\), with \(v = 150\) yards per minute, \(t=\frac{274}{150}\approx1.8\) minutes.

Answer:

a. 224 yards
b. 47 yards
c. 1.8 minutes