Answer:
8) 18π square inches
9) 2246 miles
10) 4687 miles
Step-by-step explanation:
To find the area of the sector, we can use the formula for the area of a sector of a circle:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a Sector}}\\\\A= \left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A$ is the area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]
In this case:
Substitute the values into the formula and solve for A:
[tex]A= \left(\dfrac{45^{\circ}}{360^{\circ}}\right) \pi \cdot (12)^2[/tex]
[tex]A= \left(\dfrac{1}{8}\right) \pi \cdot 144[/tex]
[tex]A= \dfrac{144}{8} \pi[/tex]
[tex]A= 18 \pi[/tex]
Therefore, the area of the sector is 18π square inches.
[tex]\hrulefill[/tex]
Two cities are on the same longitude. One is situated at latitude 15°20' and the other at 50°12'. The radius of the earth is given as about 3690 miles.
To find the distance between the two cities, we can use the arc length of a circle formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Arc length}}\\\\s= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$s$ is the arc length.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]
In this case, the central angle (θ) is the difference between the two angles.
Begin by converting both angles to decimal degrees:
[tex]50^{\circ}12'=50+\dfrac{12}{60}=50.2^{\circ}[/tex]
[tex]15^{\circ}20'=15+\dfrac{20}{60}=15.\.{3}^{\circ}[/tex]
Subtract them:
[tex]50.2^{\circ}-15.\.{3}^{\circ}=34.8\.{6}^{\circ}[/tex]
Now, substitute the angle and the radius into the formula and solve for s:
[tex]s= \pi (3690)\left(\dfrac{34.8\.{6}^{\circ}}{180^{\circ}}\right)[/tex]
[tex]s=2245.50570...[/tex]
[tex]s=2246\; \sf miles[/tex]
Therefore, the distance between the two cities is 2246 miles (to the nearest mile).
[tex]\hrulefill[/tex]
Two cities are on the same longitude. One is situated at 30°16' North of the equator and the other is 42°30'15" South of the equator. The radius of the earth is given as approximately 3690 miles.
To find the distance between the two cities, we can use the arc length of a circle formula:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Arc length}}\\\\s= \pi r\left(\dfrac{\theta}{180^{\circ}}\right)\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$s$ is the arc length.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\\phantom{ww}\bullet\;\textsf{$\theta$ is the central angle in degrees.}\end{array}}[/tex]
In this case, the central angle (θ) is the absolute difference between the two angles because we are measuring the angles with respect to the equator.
Begin by converting both angles to decimal degrees:
[tex]30^{\circ}16'=30+\dfrac{16}{60}=30.2\.{6}^{\circ}[/tex]
[tex]42^{\circ}30'15''=42+\dfrac{30}{60}+\dfrac{15}{3600}=42.5041\.{6}^{\circ}[/tex]
Find the absolute difference:
[tex]\theta=|42.5041\.{6}^{\circ}-30.2\.{6}^{\circ}|[/tex]
[tex]\theta=72.7708\.{3}^{\circ}[/tex]
Now, substitute the angle and the radius into the formula and solve for s:
[tex]s= \pi (3690)\left(\dfrac{72.7708\.{3}^{\circ}}{180^{\circ}}\right)[/tex]
[tex]s=4686.63446...[/tex]
[tex]s=4687\; \sf miles[/tex]
Therefore, the distance between the two cities is 4687 miles (to the nearest mile).
