Two math students erect a sun "shade" on the beach (similar to a lean too). This "shade" is rectangular in shape with dimensions 1.5 m long and 2 m wide, and it makes an angle of 60° with the ground. When the sun's rays cast down on this "shade" there will be an area of shade made on the ground (also a rectangle). What is the area of shade that the students will have to sit in at 12 noon ? (that is, what is the projection of the shade onto the ground)? (Assume the sun’s rays are shining directly down).
+26367 Two math students erect a sun "shade" on the beach (similar to a lean too). This "shade" is rectangular in shape with dimensions 1.5 m long and 2 m wide, and it makes an angle of 60° with the ground. When the sun's rays cast down on this "shade" there will be an area of shade made on the ground (also a rectangle). What is the area of shade that the students will have to sit in at 12 noon ? (that is, what is the projection of the shade onto the ground)? (Assume the sun’s rays are shining directly down).
$$\vec{a}=1.5\cdot
\begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\qquad
\vec{b} = \begin{pmatrix} 0\\2 \end{pmatrix}\\\\
\begin{array}{lll}
\text{area}=\left|\vec{a}_{projection}\times\vec{b} \right|
& \qquad \vec{a}_{projection} = \left(\vec{a}\cdot \vec{e}_x\right)\cdot \vec{e}_x
& \qquad \vec{e}_x = \begin{pmatrix} 1\\0 \end{pmatrix}\\\\
& \qquad \vec{a}_{projection} =\left( 1.5\cdot \begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\cdot \begin{pmatrix} 1\\0 \end{pmatrix}\right)\cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \left( 1.5\cdot \cos{(60\ensurement{^{\circ}} )} +0\right) \cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix} \\\\
\text{area}=\left|\begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix}\times\begin{pmatrix} 0\\2 \end{pmatrix} \right| \\\\
\text{area}=2\cdot 1.5\cdot \cos{ (60\ensurement{^{\circ}} ) }
\qquad | \qquad \cos{ (60\ensurement{^{\circ}} ) } = \dfrac{1}{2}\\\\
\text{area}=2\cdot 1.5\cdot \dfrac{1}{2}\\\\
\mathbf{\text{area}=1.5 ~\mathrm{m^2}}\\\\
\end{array}$$
I will make an assumption that the tarp is 1.5 m TALL and 2 m WIDE.
One side of the shadow will remain the same if the sun is shining directly down: 2m
To calculate the OTHER side of this shadow rectangle,
The hypotenuse of the triangle is 1.5 m . We need the cosine of 60 dgrees times the hypotenuse to calculate the other side of the shadow
COS(60) * 1.5 =.75
So you have a rectangle that is 2m x .75m Area = 2 * .75 = 1.5 sq m
+26367 Two math students erect a sun "shade" on the beach (similar to a lean too). This "shade" is rectangular in shape with dimensions 1.5 m long and 2 m wide, and it makes an angle of 60° with the ground. When the sun's rays cast down on this "shade" there will be an area of shade made on the ground (also a rectangle). What is the area of shade that the students will have to sit in at 12 noon ? (that is, what is the projection of the shade onto the ground)? (Assume the sun’s rays are shining directly down).
$$\vec{a}=1.5\cdot
\begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\qquad
\vec{b} = \begin{pmatrix} 0\\2 \end{pmatrix}\\\\
\begin{array}{lll}
\text{area}=\left|\vec{a}_{projection}\times\vec{b} \right|
& \qquad \vec{a}_{projection} = \left(\vec{a}\cdot \vec{e}_x\right)\cdot \vec{e}_x
& \qquad \vec{e}_x = \begin{pmatrix} 1\\0 \end{pmatrix}\\\\
& \qquad \vec{a}_{projection} =\left( 1.5\cdot \begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\cdot \begin{pmatrix} 1\\0 \end{pmatrix}\right)\cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \left( 1.5\cdot \cos{(60\ensurement{^{\circ}} )} +0\right) \cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix} \\\\
\text{area}=\left|\begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix}\times\begin{pmatrix} 0\\2 \end{pmatrix} \right| \\\\
\text{area}=2\cdot 1.5\cdot \cos{ (60\ensurement{^{\circ}} ) }
\qquad | \qquad \cos{ (60\ensurement{^{\circ}} ) } = \dfrac{1}{2}\\\\
\text{area}=2\cdot 1.5\cdot \dfrac{1}{2}\\\\
\mathbf{\text{area}=1.5 ~\mathrm{m^2}}\\\\
\end{array}$$
