Question

I need help with my pre-calculus homework, please show me how to solve them step by step if possible. The image of the problem is attached. These are 2 parts of the same question.

Answer 1

We are given the following triangle:

We need to determine the area of the triangle. To do that we need to determine sides "a" and "b". We will use the sine law to determine the side "b":

`(b)/(sin107)=(98)/(sin48)`

Now, we multiply both sides by "sin107":

`b=sin(107)(98ft)/(sin(48))`

Solving the operations:

`b=126.11ft`

Now, before determining side "a" we will determine the angle "x" that is opposed to "a". To do that we will use the fact that the sum of the interior angles of a triangle is 180, therefore:

`107+48+x=180`

Adding the values:

`155+x=180`

Now, we subtract 155 from both sides:

`\begin{gathered} x=180-155 \\ x=25 \end{gathered}`

Therefore, the angle opposite to "a" is 25 degrees. Now, we apply the sine law:

`(a)/(sin(25))=(98)/(sin(48))`

Now, we multiply both sides by "sin(25)":

`a=sin(25)(98)/(sin(48))`

Solving the operations:

`a=55.73ft`

Now, we determine the area using the following formula:

`A=√(s(s-a)(s-b)(s-c))`

Where:

`s=(a+b+c)/(2)`

Now, we determine the value of "s":

`s=(55.73ft+126.11ft+98ft)/(2)`

Solving the operation:

`s=139.92ft`

Now, we substitute the value in the formula for the area:

`A=√((139.92ft)(139.92ft-55.73ft)(139.92ft-126.11ft)(139.92ft-98ft))`

Solving the operations:

`A=2611.43ft^2`

Now, since the search party can cover 300 ft^2/h we can use a rule of 3 to determine the number of hours it takes them to cover 2611.43 ft^2:

`\begin{gathered} 300ft^2\rightarrow1h \\ 2611.43ft^2\rightarrow x \end{gathered}`

Now, we cross multiply:

`(300ft^2)(x)=(1h)(2611.43ft^2)`

Now, we divide both sides by 300ft^2:

`x=((1h)(2611.43ft^2))/((300ft^2))`

Solving the operations:

`x=8.7h`

Therefore, it takes 8.7 hours to cover the area. Therefore, the search party won't be able to conclude before the sun goes down.