Needs Bulb

Needs Bulb
Student has light bulb 100 cm from screen and needs image of bulb with converging lens of focal length 15 cm?

A student is working in an optics lab and has a light bulb which is 100 cm from a screen. She needs to make a real image of the bulb on the screen and she has a converging lens with a focal length of +15 cm. Which two places can she put the lens? And what are the object and image distances for both setups?

R1 = obj. dist., R2 = img. dist.
RT = R1+R2 = 1 m.
F = 0.15 m.
From the thin-lens equation (ref.),
1/F = 1/R1+1/R2 ==> 1/F = 1/R1+1/(RT-R1) ==> R1(RT-R1)/F = RT-R1+R1 = RT ==>
quadratic: -1/F*R1^2 + RT/F*R1 - RT = 0.
R1 = 0.183772233983162
R2 = 0.816227766016838
Since F>0 the image is real so R1 and R2 values can be swapped.
Thus the lens can be 0.183772233983162 m or 0.816227766016838 m from the lamp.
Check:
0.183772233983162 m+0.816227766016838 m = 1 m
f = 1/(1/0.183772233983162 + 1/0.816227766016838) = 0.15 m

Hot bulb engines and old machines, old truck