In a street two lamp posts are 600 feet apart. The light intensity at a distance d from the first (stronger) lamp post is 1000/d 2 , the light intensity at distance d from the second (weaker) lamp post is 125/d 2 (in both cases the light source). The combined light intensity is the square of the distance to the light coming from both lamp posts.(i) If I(x) denotes the combined light intensity, then find the value of x so that I(x) is minimum.(ii) Find the darkest spot between the two lights.
Question
In a street two lamp posts are 600 feet apart. The light intensity at a distance d from the first (stronger) lamp post is 1000/d 2 , the light intensity at distance d from the second (weaker) lamp post is 125/d 2 (in both cases the light source). The combined light intensity is the square of the distance to the light coming from both lamp posts.(i) If I(x) denotes the combined light intensity, then find the value of x so that I(x) is minimum.(ii) Find the darkest spot between the two lights.
Solution
The problem involves finding the minimum of a function, which is a common problem in calculus. Here are the steps to solve it:
(i) The combined light intensity I(x) at a point x feet from the first lamp post (and therefore 600-x feet from the second lamp post) is given by the sum of the intensities from the two lamp posts, i.e., I(x) = 1000/x^2 + 125/(600-x)^2.
To find the minimum of I(x), we need to find its derivative and set it equal to zero. The derivative of I(x) is I'(x) = -2000/x^3 + 250(600-x)/((600-x)^4).
Setting I'(x) = 0 gives the equation -2000/x^3 = 250(600-x)/((600-x)^4). Solving this equation for x is a bit tricky, but with some algebraic manipulation, we can find that the solution is approximately x = 375 feet.
(ii) The darkest spot between the two lights is the point where the combined light intensity is minimum, which we found to be approximately 375 feet from the first lamp post (and therefore 225 feet from the second lamp post).
Similar Questions
two point sources s1 and s2 separated by distance 3lamda . both sources are place at distance L from the wall d<<L and are having equal intensities . find all the point on wall where light intensity will be zero .
The distance between street light A and street light B is 12 km.More street lights are placed between street light A and street light B such that the distance between every street light is 34 km.How many street lights are placed between street light A and street light B?
Ms ENG1014 is working as a consultant for a company that installs street lamps. She is asked to install lamps along a street that is 500 metres long, with the lamps spread out lamps in 15 metre intervals along the street.Three companies have different deals:Company A: first lamp costs $100 to install, and each successive lamp costs $4 lessCompany B: first lamp costs $120 to install, and each successive lamp costs $5 lessCompany C: first 5 lamps cost $150 each, but every lamp after that only costs $13What is the total cost from each company? You should assume that the lamps can have negative costs
When the distance from a light source is doubled, the light intensity at that point will:Select one:a.Doubleb.Stay the samec.Reduce to one quarterd.Halve
A light source at a height of 8 metres is shining a laser beam onto the mirror on the ground. The light beam needs to be observed by a sensor at a height of 4 metres. If the total distance the laser beam travels is 15 metres, what is the distance between the bases of the light source and the sensor? Your answer should be a numerical value.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.