Math / Geometry

QuestionA painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure 12.3 cm and 16.4 cm . She knows that the angle between the two sides is $116^{\circ}$ . What is the area of the surface she plans to cover with the coating?
The surface area is $\square$ $\mathrm{cm}^{2}$ (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as needed.)

Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle given two sides and the angle between them. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember that the area formula uses the sine of the angle, not the cosine or tangent.

STEP 2

1. Set up the Area Formula
2. Calculate the Area

STEP 3

The area of a triangle given two sides $a$ and $b$ and the included angle $C$ is given by: $A = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)$ This formula is super handy when you have this specific information about a triangle.
It's like a shortcut to the area!

STEP 4

In our problem, we have $a = 12.3$ cm, $b = 16.4$ cm, and $C = 116^{\circ}$ .
Let's plug these values into our awesome formula: $A = \frac{1}{2} \cdot 12.3 \cdot 16.4 \cdot \sin(116^{\circ})$ See how nicely everything fits?
We're almost there!

STEP 5

First, let's find the sine of the angle.
Make sure your calculator is in degree mode! $\sin(116^{\circ}) \approx \mathbf{0.8988}$ Keep this value handy; we'll use it in the next step.

STEP 6

Now, we can multiply everything together: $A = \frac{1}{2} \cdot 12.3 \cdot 16.4 \cdot \mathbf{0.8988}$ $A \approx \frac{1}{2} \cdot 201.72 \cdot \mathbf{0.8988}$ $A \approx 100.86 \cdot \mathbf{0.8988}$ $A \approx \mathbf{90.6312}$ So, the area is approximately $\mathbf{90.6312}$ $\text{cm}^2$ .

STEP 7

The problem asks us to round to the nearest tenth.
Looking at our result, the tenths place is $6$ , and the next digit is $3$ , which is less than $5$ .
So we round down: $A \approx \mathbf{90.6}$

STEP 8

The area of the triangular metal plate is approximately $\mathbf{90.6}$ $\text{cm}^2$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle given two sides and the angle between them. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember that the area formula uses the sine of the angle, not the cosine or tangent.

STEP 2

1. Set up the Area Formula
2. Calculate the Area

STEP 3

The area of a triangle given two sides $a$ and $b$ and the included angle $C$ is given by: $A = \frac{1}{2} \cdot a \cdot b \cdot \sin(C)$ This formula is super handy when you have this specific information about a triangle.
It's like a shortcut to the area!

STEP 4

In our problem, we have $a = 12.3$ cm, $b = 16.4$ cm, and $C = 116^{\circ}$ .
Let's plug these values into our awesome formula: $A = \frac{1}{2} \cdot 12.3 \cdot 16.4 \cdot \sin(116^{\circ})$ See how nicely everything fits?
We're almost there!

STEP 5

First, let's find the sine of the angle.
Make sure your calculator is in degree mode! $\sin(116^{\circ}) \approx \mathbf{0.8988}$ Keep this value handy; we'll use it in the next step.

STEP 6

Now, we can multiply everything together: $A = \frac{1}{2} \cdot 12.3 \cdot 16.4 \cdot \mathbf{0.8988}$ $A \approx \frac{1}{2} \cdot 201.72 \cdot \mathbf{0.8988}$ $A \approx 100.86 \cdot \mathbf{0.8988}$ $A \approx \mathbf{90.6312}$ So, the area is approximately $\mathbf{90.6312}$ $\text{cm}^2$ .

STEP 7

The problem asks us to round to the nearest tenth.
Looking at our result, the tenths place is $6$ , and the next digit is $3$ , which is less than $5$ .
So we round down: $A \approx \mathbf{90.6}$

STEP 8

The area of the triangular metal plate is approximately $\mathbf{90.6}$ $\text{cm}^2$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the area of a triangle given two sides and the angle between them. Watch out! Make sure your calculator is in degree mode, not radians!Also, remember that the area formula uses the *sine* of the angle, not the cosine or tangent.

STEP 2

1. Set up the Area Formula2. Calculate the Area

STEP 3

STEP 4

STEP 5

First, let's find the sine of the angle.Make sure your calculator is in degree mode! sin⁡(116∘)≈0.8988 \sin(116^{\circ}) \approx \mathbf{0.8988} sin(116∘)≈0.8988 Keep this value handy; we'll use it in the next step.

STEP 6

STEP 7

The problem asks us to round to the nearest tenth.Looking at our result, the tenths place is 666, and the next digit is 333, which is less than 555.So we round down: A≈90.6 A \approx \mathbf{90.6} A≈90.6

STEP 8

The area of the triangular metal plate is approximately 90.6\mathbf{90.6}90.6 cm2\text{cm}^2cm2.

Get Studdy

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.

What is this asking? We need to find the area of a triangle given two sides and the angle between them. Watch out! Make sure your calculator is in degree mode, not radians!
Also, remember that the area formula uses the sine of the angle, not the cosine or tangent.

1. Set up the Area Formula
2. Calculate the Area

First, let's find the sine of the angle.
Make sure your calculator is in degree mode! $\sin(116^{\circ}) \approx \mathbf{0.8988}$ Keep this value handy; we'll use it in the next step.

The problem asks us to round to the nearest tenth.
Looking at our result, the tenths place is $6$ , and the next digit is $3$ , which is less than $5$ .
So we round down: $A \approx \mathbf{90.6}$

The area of the triangular metal plate is approximately $\mathbf{90.6}$ $\text{cm}^2$ .