The distance between two coordinate points is the number of units between them.
The distance from where the kite is tied to the shadow is 3.09 feet.
The coordinates of the kite is given as:
[tex]\mathbf{(x,y) = (0,0)(3,12)}[/tex]
Calculate the slope
[tex]\mathbf{m = \frac{y_2 - y_1}{x_2 - x_1}}[/tex]
[tex]\mathbf{m = \frac{12 - 0}{3-0}}[/tex]
[tex]\mathbf{m = \frac{12}{3}}[/tex]
[tex]\mathbf{m = 4}[/tex]
The shadow is perpendicular to the kite.
So, the slope (m2) of the shadow is:
[tex]\mathbf{m_2 = -\frac 1m}[/tex]
[tex]\mathbf{m_2 = -\frac 14}[/tex]
The equation of the shadow is:
[tex]\mathbf{y = m_2(x - x_1) + y_1}[/tex]
[tex]\mathbf{y = -\frac 14(x - 3) + 12}[/tex]
[tex]\mathbf{y = -\frac 14x + \frac34 + 12}[/tex]
Set x = 0, to calculate the coordinate of the shadow
[tex]\mathbf{y = -\frac 14(0)+ \frac34 + 12}[/tex]
[tex]\mathbf{y = 0+ \frac34 + 12}[/tex]
[tex]\mathbf{y = \frac{51}4}[/tex]
So, the coordinate of the shadow is; [tex]\mathbf{(0,\frac{51}{4})}[/tex]
The distance is the calculated using:
[tex]\mathbf{d = \sqrt{(x_2 - x_1)^2 + (y_2 -y_1)^2}}[/tex]
This gives
[tex]\mathbf{d = \sqrt{(0 - 3)^2 + (51/4 -12)^2}}[/tex]
[tex]\mathbf{d = \sqrt{(0 - 3)^2 + (12.75 -12)^2}}[/tex]
[tex]\mathbf{d = \sqrt{9 + 0.5625}}[/tex]
[tex]\mathbf{d = \sqrt{9.5625}}[/tex]
[tex]\mathbf{d = 3.09}[/tex]
Hence, the distance from where the kite is tied to the shadow is 3.09 feet.
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