two homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved…

two homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. sally can jog 7 miles per hour along the paved path, but only 5 miles per hour in the sand. because of a river between the two houses, it is necessary to jog on the sand to the path, continue on the path, and then jog on the sand to get from one house to the other. complete parts (a) through (g).\na. $t = \\frac{2\\csc\\theta}{5}+\\frac{8 - 2\\cot\\theta}{7}$\nb. $t = \\frac{\\csc\\theta}{5}+\\frac{8 - \\cot\\theta}{7}$\nc. $t = \\frac{2\\sin\\theta}{5}+\\frac{8 - 2\\tan\\theta}{7}$\nd. $t = \\frac{\\sin\\theta}{5}+\\frac{8 - \\tan\\theta}{7}$\n(b) calculate the time t for $\\theta = 30^{\\circ}$. how long is sally on the paved path?\n\\(\\square\\) hours\n(do not round until the final answer. then round to two decimal places as needed.)

two homes are located 8 miles apart on a straight stretch of beach, each a distance of 1 mile from a paved path that parallels the ocean. sally can jog 7 miles per hour along the paved path, but only 5 miles per hour in the sand. because of a river between the two houses, it is necessary to jog on the sand to the path, continue on the path, and then jog on the sand to get from one house to the other. complete parts (a) through (g).\na. $t = \\frac{2\\csc\\theta}{5}+\\frac{8 - 2\\cot\\theta}{7}$\nb. $t = \\frac{\\csc\\theta}{5}+\\frac{8 - \\cot\\theta}{7}$\nc. $t = \\frac{2\\sin\\theta}{5}+\\frac{8 - 2\\tan\\theta}{7}$\nd. $t = \\frac{\\sin\\theta}{5}+\\frac{8 - \\tan\\theta}{7}$\n(b) calculate the time t for $\\theta = 30^{\\circ}$. how long is sally on the paved path?\n\\(\\square\\) hours\n(do not round until the final answer. then round to two decimal places as needed.)

Answer

Explanation:

Step1: Recall trigonometric - function values for $\theta = 30^{\circ}$

$\csc\theta=\frac{1}{\sin\theta}$, $\cot\theta=\frac{\cos\theta}{\sin\theta}$. When $\theta = 30^{\circ}$, $\sin\theta=\frac{1}{2}$, $\cos\theta=\frac{\sqrt{3}}{2}$, so $\csc30^{\circ}=2$, $\cot30^{\circ}=\sqrt{3}$.

Step2: Substitute into the time - formula (from part A)

The formula from part A is $T=\frac{2\csc\theta}{5}+\frac{8 - 2\cot\theta}{7}$. Substitute $\csc30^{\circ}=2$ and $\cot30^{\circ}=\sqrt{3}$ into the formula: [ \begin{align*} T&=\frac{2\times2}{5}+\frac{8-2\sqrt{3}}{7}\ &=\frac{4}{5}+\frac{8 - 2\sqrt{3}}{7}\ &=\frac{4\times7}{5\times7}+\frac{(8 - 2\sqrt{3})\times5}{7\times5}\ &=\frac{28}{35}+\frac{40-10\sqrt{3}}{35}\ &=\frac{28 + 40-10\sqrt{3}}{35}\ &=\frac{68-10\sqrt{3}}{35}\approx\frac{68 - 10\times1.732}{35}\ &=\frac{68 - 17.32}{35}\ &=\frac{50.68}{35}\approx1.45 \end{align*} ]

Step3: Find the length on the paved path

The length of the paved - path is $8 - 2\cot\theta$. Substitute $\theta = 30^{\circ}$ and $\cot30^{\circ}=\sqrt{3}$: $8-2\sqrt{3}\approx8 - 2\times1.732=8 - 3.464 = 4.54$ miles.

Answer:

1.45 hours, 4.54 miles