Question

The length of the diagonal of a Rectangle is 14cm,and it forms a 30 degree angle in one corner of the rectangle.What is the area of the rectangle.(A=LxW)Just number 20

Answer 1

`\text{Area}=84.87(cm^2)`

Step-by-step explanation

Step 1

draw the rectangle

here we have a rigth triangle,then

Let

`\begin{gathered} hypotenuse=14 \\ agle=30\text{ \degree} \\ \text{adjacent side= length= l} \end{gathered}`

so, we need a function that relates those values

`\cos \Theta=\frac{adjacent\text{ side}}{\text{hypotenuse}}`

replace and solve for length

`\begin{gathered} \cos \Theta=\frac{adjacent\text{ side}}{\text{hypotenuse}} \\ \text{hypotenuse}\cdot\cos \Theta=adjacent\text{ side} \\ 14\text{ cm }\cdot\cos 30=l \\ 12.12435\text{ cm=l} \end{gathered}`

Step 2

width

similarity, we need a function that relates

`\sin \text{ }\Theta=\frac{opposite\text{ side}}{\text{hypotenuse}}`

let

`\text{opposite side= width=w}`

replace and solve for w

`\begin{gathered} \sin \text{ }\Theta=\frac{opposite\text{ side}}{\text{hypotenuse}} \\ \text{hypotenuse}\cdot\sin \Theta=opposite\text{ side} \\ 14\text{ cm }\cdot\sin \text{ 30=w} \\ 7cm=w \end{gathered}`

Step 3

finally, the area of a rectangle is given by

`\begin{gathered} \text{Area}=\text{ length }\cdot width \\ \text{replacing} \\ \text{Area}=(12.12\cdot7)(cm^2) \\ \text{Area}=84.87(cm^2) \end{gathered}`

therefore, the answer is

`\text{Area}=84.87(cm^2)`

I hope this helps you