An object of height 10cm placed at a distance of 15 from the pole produced an image is twice the height of the object. Calculate the length of the camera and distance of the image from the pinhole camera
Let the length of the camera be denoted as l, and the distance of the image from the pinhole camera be denoted as d.
According to the thin lens formula, 1/f = 1/v - 1/u, where f is the focal length, v is the distance of the image from the lens, and u is the distance of the object from the lens.
Given that the height of the object is 10 cm and it produces an image that is twice its height, the image height would be 2 * 10 cm = 20 cm.
Using similar triangles, we can set up the equation h/v = h/u, where h is the object height, v is the image distance from the pinhole camera, and u is the object distance from the pinhole camera.
Substituting the given values, (20 cm) / v = (10 cm) / (15 cm + l).
We can rewrite the equation as: v = (20 cm) / ((10 cm) / (15 cm + l)).
Since the image distance is d, we have v = d + l.
Substituting this into the previous equation, we get: d + l = (20 cm) / ((10 cm) / (15 cm + l)).
Simplifying the equation further, we have (15 cm + l) * (d + l) = 20 cm.
Expanding the equation, we get: 15d + 15l + l*d + l^2 = 20 cm.
Rearranging the equation and combining like terms, we obtain: l^2 + (d + 15)l + (15d - 20) cm = 0.
Using the quadratic formula, l = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = (d + 15), and c = (15d - 20).
Calculating the discriminant (b^2 - 4ac), we have discriminant = (d + 15)^2 - (4)(1)(15d - 20) = d^2 + 30d + 225 - 60d + 80 = d^2 - 30d + 305.
Since we want the length of the camera to be positive, the discriminant should be greater than or equal to zero.
Therefore, d^2 - 30d + 305 ≥ 0.
We can solve this inequality by factoring, completing the square, or using the quadratic formula.
If we use the quadratic formula and solve for d, we get: d = (30 ± √(30^2 - 4(1)(305))) / 2 = (30 ± √(900 - 1220)) / 2 = (30 ± √(-320)) / 2.
Since the square root of a negative number is not a real number, the quadratic formula does not yield a valid solution.
Therefore, there is no real value for d that satisfies the given conditions.