His eyes are 1.76 m above the floor, and the top of his head is 0.15 m higher. Find the height (in m) above the floor of the top and bottom of the smallest mirror in which he can see both the top of his head and his feet.

The height (in m) above the floor of the top and bottom of the smallest mirror in which he can see both the top of his head and his feet is;

1.835 m and 0.88 m.

Explanation:

Here we have the total height of the man as

1.76 + 0.15 = 1.91 m

The mirror is positioned such that the person can see both the top of his head and his feet

We have the eyes are 0.15 m below the top of the head, therefore by the law of reflection, the incident and reflected angle must be equal.

Hence, the light from the top of his head and then reflected to his eyes forms a isosceles triangle, with the base being the distance of the eye to the top of his head and the top of the triangle is on the mirror.

The height of the mirror is then

1.91 - 0.15/2 = 1.835 m

Similarly, the distance from the eye to the feet is 1.76, therefore, the base of the mirror is positioned at 1.76/2 or 0.88 m above the ground.

Bottom of mirror to ground = 0.88m

Top of mirror to ground = 1.835m

Explanation:

From the image attached;

- A D is the distance between his eyes and the ground which is 1.76m

- A F is the distance between his eyes and the top of his head which is 0.15

-G D is the distance from the top of the mirror to the ground

-B D is distance from the bottom of mirror to the ground

From law of refraction, θi = θr and from trigonometric ratios, we can say that;

A B/B C = B D/B C

Thus, BC will cancel out and we have;

A B = B D

Now, from the diagram,

A D = A B + B D

Thus, since A B = B D

A D = B D + B D = 2B D

From the question, A D = 1.76,Thus

B D = A D/2 = 1.76/2 = 0.88m

Now,for the top of the mirror;

G D = G B + B D

As said before From law of refraction, θi = θr and from trigonometric ratios, we can say that;

G A/G E = G F/G E

G E will cancel out and we have;

G A = G F

Now, from the diagram, A F = G F + G A

Now we already know A F = 0.15m and since G A = G F,

Thus, 0.15 = G A + G A = 2G A

G A = 0.15/2 = 0.075m

From the diagram, G A + A D will give us the height of the top of the mirror above the ground.

Thus,

G D = G A + A D = 0.075 + 1.76 = 1.835m

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