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The equation to model the situation is [tex]\mathbf{y = \dfrac{k}{x^2}}[/tex]. The constant for the variation is 2250.
The intensity of light from a lantern varies inversely to the square of the distance from the lantern.
From the given information:
Then:
[tex]\mathbf{y \alpha \dfrac{1}{x^2} }[/tex]
[tex]\mathbf{y = \dfrac{k}{x^2} }[/tex] here, k = constant.
2.
If y = 90 W/m² when the distance x = 5m
Then:
[tex]\mathbf{90 = \dfrac{k}{(5)^2}}[/tex]
k = 90 × 25
k = 2250
c.
The equation to model the situation by using the constant variation is:
[tex]\mathbf{y = \dfrac{2250}{x^2}}[/tex]
d.
If the light intensity y = 40, then x is determined as:
[tex]\mathbf{40 = \dfrac{2250}{x^2}}[/tex]
[tex]\mathbf{x = \sqrt{\dfrac{2250}{40}}}[/tex]
x = 7.5 m
e.
The light is needed in (225 × 1000)m = 225000 km of illumination.
f.
The lantern required for the new light estimation is:
y = 2250/225000
y = 0.01 intensity
Therefore, we can conclude that to get an intensity of 1 W/m², we need to put 100 lanterns.
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