ok the question is:
consider a concave mirror with a focal length of 13.72 cm
find the image distance when the object distance is 6.86 cm.
ok so i did
1/6.86 + 1/q = 1/13.72
so 1/q = -.0728862974
is this right?
Yes. Converting to the image distance, that means q = -13.9 cm
thanks :]
To solve the problem, you correctly used the mirror equation:
1/f = 1/p + 1/q
where f is the focal length, p is the object distance, and q is the image distance.
Now, let's substitute the given values into the equation:
1/13.72 = 1/6.86 + 1/q
To simplify the equation, we can multiply both sides by a common denominator to eliminate the fractions. In this case, the common denominator is 6.86q:
6.86q/13.72 = 6.86q/6.86 + 13.72/6.86q
Now, let's simplify further:
6.86q/13.72 = 1 + 13.72/6.86q
Now, subtract 1 from both sides:
6.86q/13.72 - 1 = 1 + 13.72/6.86q - 1
6.86q/13.72 - 1 = 13.72/6.86q
To make both sides of the equation have the same denominator, we can multiply the value on the left side by q/q and the value on the right side by 13.72/13.72:
(6.86q*q)/(13.72*q) - q = (13.72*13.72)/(6.86q*13.72)
Now, simplify further:
6.86q*q - 13.72q = 189.1424/q
At this point, we have a quadratic equation. Let's rearrange it into standard form:
6.86q^2 - 13.72q - 189.1424 = 0
To solve this quadratic equation, you can use the quadratic formula:
q = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 6.86, b = -13.72, and c = -189.1424. Let's substitute these values into the formula:
q = (-(-13.72) ± √((-13.72)^2 - 4(6.86)(-189.1424))) / (2(6.86))
Simplifying further:
q = (13.72 ± √(188.2224 + 5203.589376)) / 13.72
q = (13.72 ± √(5391.811776)) / 13.72
q = (13.72 ± 73.4392) / 13.72
Now, we have two possible values for q:
q1 = (13.72 + 73.4392) / 13.72
q1 = 87.1592 / 13.72
q1 ≈ 6.34 cm
q2 = (13.72 - 73.4392) / 13.72
q2 = -59.7192 / 13.72
q2 ≈ -4.35 cm
Since concave mirrors produce real, inverted images on the same side as the object, we can conclude that the image distance is approximately 6.34 cm when the object distance is 6.86 cm.