Math / Trigonometry

Questionfor each problem below, draw and label the associated triangle. Next, determine if you will need to use the sine law or cosine law to find the unknowns (you may be able to use both). solve for the unknown(s) and show your work. Finally write a convulsing statement that answers the question. Round your answers to one decimal place. 1.A swearer pipe for a new subdivision has to be laid underground. A connection is, Ade to the main service pipe are either end of the 4.8 km stretch of road. One pipe, 2.5 km long, makes an angle of 72 degrees at one end of the road. What is the length go a second pipe that will connect the first pipe to the end of the road?

Studdy Solution

STEP 1

What is this asking? Find the length of a pipe needed to connect two other pipes, forming a triangle, knowing one side is 4.8 km, another is 2.5 km, and the angle opposite the unknown side is $72^\circ$ . Watch out! Make sure to use the correct trigonometric law (sine law) and watch out for the units!

STEP 2

1. Draw the Triangle
2. Choose the Right Tool
3. Calculate the Unknown Side

STEP 3

Alright, let's picture this!
We've got a road that's 4.8 km long.
That's one side of our triangle.
Then we have a pipe, 2.5 km long, connected to one end of the road.
This pipe makes a 72-degree angle with the road.
We need to find the length of the second pipe that connects the other end of the road to the first pipe.
So, our triangle has sides of 4.8 km and 2.5 km, and an angle of $72^\circ$ .

STEP 4

Let's draw this triangle!
We'll label the road as side $c$ ( $4.8$ km), the first pipe as side $a$ ( $2.5$ km), and the unknown second pipe as side $b$ .
The angle opposite side $b$ (the angle between the road and the first pipe) is $C = 72^\circ$ .

STEP 5

We know two sides ( $a$ and $c$ ) and the angle opposite one of them ( $C$ ).
This screams sine law!
Remember, the sine law says:
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

STEP 6

We want to find side $b$ , and we know $a$ , $c$ , and $C$ .
So, we'll use this part of the sine law:
$\frac{\sin B}{b} = \frac{\sin C}{c}$

STEP 7

Let's plug in what we know: $a = 2.5$ km, $c = 4.8$ km, and $C = 72^\circ$ .
We're looking for $b$ .
We don't know $B$ yet, but we'll get there!
$\frac{\sin 72^\circ}{4.8} = \frac{\sin B}{b}$

STEP 8

First, let's isolate $\sin B$ : $\sin B = \frac{b \cdot \sin 72^\circ}{4.8}$ We also know from the sine rule that: $\frac{\sin A}{a} = \frac{\sin C}{c}$ So: $\sin A = \frac{a \cdot \sin C}{c} = \frac{2.5 \cdot \sin 72^\circ}{4.8} \approx \frac{2.5 \cdot 0.951}{4.8} \approx \frac{2.3775}{4.8} \approx 0.495$ Therefore: $A = \arcsin(0.495) \approx 29.7^\circ$

STEP 9

Since the angles in a triangle add up to $180^\circ$ , we can find angle $B$ : $B = 180^\circ - A - C = 180^\circ - 29.7^\circ - 72^\circ = 78.3^\circ$

STEP 10

Now we can use the sine rule to find $b$ : $\frac{b}{\sin B} = \frac{c}{\sin C}$ $b = \frac{c \cdot \sin B}{\sin C} = \frac{4.8 \cdot \sin 78.3^\circ}{\sin 72^\circ} \approx \frac{4.8 \cdot 0.979}{0.951} \approx \frac{4.6992}{0.951} \approx 4.9$ So, $b \approx 4.9$ km.

STEP 11

The length of the second pipe needs to be approximately 4.9 km.

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Studdy Solution

STEP 1

STEP 2

1. Draw the Triangle
2. Choose the Right Tool
3. Calculate the Unknown Side

STEP 3

STEP 4

Let's draw this triangle!
We'll label the road as side $c$ ( $4.8$ km), the first pipe as side $a$ ( $2.5$ km), and the unknown second pipe as side $b$ .
The angle opposite side $b$ (the angle between the road and the first pipe) is $C = 72^\circ$ .

STEP 5

We know two sides ( $a$ and $c$ ) and the angle opposite one of them ( $C$ ).
This screams sine law!
Remember, the sine law says:
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

STEP 6

We want to find side $b$ , and we know $a$ , $c$ , and $C$ .
So, we'll use this part of the sine law:
$\frac{\sin B}{b} = \frac{\sin C}{c}$

STEP 7

Let's plug in what we know: $a = 2.5$ km, $c = 4.8$ km, and $C = 72^\circ$ .
We're looking for $b$ .
We don't know $B$ yet, but we'll get there!
$\frac{\sin 72^\circ}{4.8} = \frac{\sin B}{b}$

STEP 8

STEP 9

Since the angles in a triangle add up to $180^\circ$ , we can find angle $B$ : $B = 180^\circ - A - C = 180^\circ - 29.7^\circ - 72^\circ = 78.3^\circ$

STEP 10

Now we can use the sine rule to find $b$ : $\frac{b}{\sin B} = \frac{c}{\sin C}$ $b = \frac{c \cdot \sin B}{\sin C} = \frac{4.8 \cdot \sin 78.3^\circ}{\sin 72^\circ} \approx \frac{4.8 \cdot 0.979}{0.951} \approx \frac{4.6992}{0.951} \approx 4.9$ So, $b \approx 4.9$ km.

STEP 11

The length of the second pipe needs to be approximately 4.9 km.

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Studdy Solution

STEP 1

STEP 2

1. Draw the Triangle2. Choose the Right Tool3. Calculate the Unknown Side

STEP 3

STEP 4

Let's draw this triangle!We'll label the road as side ccc (4.84.84.8 km), the first pipe as side aaa (2.52.52.5 km), and the unknown second pipe as side bbb.The angle opposite side bbb (the angle between the road and the first pipe) is C=72∘C = 72^\circC=72∘.

STEP 5

We know two sides (aaa and ccc) and the angle opposite one of them (CCC).This screams **sine law**!Remember, the sine law says:sin⁡Aa=sin⁡Bb=sin⁡Cc \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} asinA​=bsinB​=csinC​

STEP 6

We want to find side bbb, and we know aaa, ccc, and CCC.So, we'll use this part of the sine law:sin⁡Bb=sin⁡Cc \frac{\sin B}{b} = \frac{\sin C}{c} bsinB​=csinC​

STEP 7

Let's plug in what we know: a=2.5a = 2.5a=2.5 km, c=4.8c = 4.8c=4.8 km, and C=72∘C = 72^\circC=72∘.We're looking for bbb.We don't know BBB yet, but we'll get there!sin⁡72∘4.8=sin⁡Bb \frac{\sin 72^\circ}{4.8} = \frac{\sin B}{b} 4.8sin72∘​=bsinB​

STEP 8

STEP 9

Since the angles in a triangle add up to 180∘180^\circ180∘, we can find angle BBB: B=180∘−A−C=180∘−29.7∘−72∘=78.3∘ B = 180^\circ - A - C = 180^\circ - 29.7^\circ - 72^\circ = 78.3^\circ B=180∘−A−C=180∘−29.7∘−72∘=78.3∘

STEP 10

STEP 11

The length of the second pipe needs to be approximately **4.9 km**.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Draw the Triangle
2. Choose the Right Tool
3. Calculate the Unknown Side

Let's draw this triangle!
We'll label the road as side $c$ ( $4.8$ km), the first pipe as side $a$ ( $2.5$ km), and the unknown second pipe as side $b$ .
The angle opposite side $b$ (the angle between the road and the first pipe) is $C = 72^\circ$ .

We know two sides ( $a$ and $c$ ) and the angle opposite one of them ( $C$ ).
This screams sine law!
Remember, the sine law says:
$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

We want to find side $b$ , and we know $a$ , $c$ , and $C$ .
So, we'll use this part of the sine law:
$\frac{\sin B}{b} = \frac{\sin C}{c}$

Let's plug in what we know: $a = 2.5$ km, $c = 4.8$ km, and $C = 72^\circ$ .
We're looking for $b$ .
We don't know $B$ yet, but we'll get there!
$\frac{\sin 72^\circ}{4.8} = \frac{\sin B}{b}$

Since the angles in a triangle add up to $180^\circ$ , we can find angle $B$ : $B = 180^\circ - A - C = 180^\circ - 29.7^\circ - 72^\circ = 78.3^\circ$

The length of the second pipe needs to be approximately 4.9 km.