A lamp located with its base at (3,0) has an unspecified height and casts a shadow on the ellipse 𝑥^2+4𝑦^2 = 5 such that (−5,0) is on the edge of the shadow. How tall is the lamp?
(-5,0) is not on the ellipse.
Not sure what's going on here.
I don't know either. It was a homework problem and I was confused.
yeah. And it casts a shadow, but a shadow of what?
is there some object somewhere between (3,0) and (-5,0)?
If so, how tall is the object?
If this is all you got, then it's a very poorly worded question.
yeah... I don't know what to do. I'll just look up the problem on the Internet to see if there's any info about it.
To find the height of the lamp, we need to determine the coordinates at which the top of the lamp meets the ellipse.
First, let's understand the problem setup. We are given that the base of the lamp is located at (3,0) and the shadow of the lamp falls on the ellipse 𝑥^2+4𝑦^2=5, with the point (-5,0) lying on the edge of the shadow.
Now, let's solve the problem step by step:
Step 1: Find the equation of the line connecting the base of the lamp and the point on the edge of the shadow.
The two coordinates we have are (3,0) and (-5,0). Since both points lie on the x-axis, the equation of the line can be written as y = 0.
Step 2: Find the points of intersection between the line and the ellipse to determine the height of the lamp.
Substitute the equation of the line into the equation of the ellipse:
x^2 + 4(0)^2 = 5
x^2 = 5
x = ±√5
So, the points of intersection are (√5, 0) and (-√5, 0).
Step 3: Calculate the distance between the base of the lamp and the top of the lamp.
The height of the lamp is equal to the y-coordinate of the points of intersection. Since the base of the lamp is located at (3,0) and one of the points of intersection is (√5,0), we can calculate the difference in y-coordinates:
Height of the lamp = y-coordinate of the point of intersection - y-coordinate of the base
= 0 - 0
= 0
Therefore, the height of the lamp is 0.