A light on top of a post is14 feet high, a 5 foot woman stands 18 feet away. how long is her shadow?

Question

A light, on top of a post, is 14 feet above the ground. At night a 5 foot woman is standing 18 feet from the post

How long is the womans shadow?

And then..

Her friend who is 6 foot tall is standing so that the tip of his shadow coincides with the tip of her shadow. How far from the post is he standing?

Answer 1

draw a picture

o*

|

|------------------o

|____18'____|__x'__

14 is the height of the lampost. 5' is the height of the woman. 18' is the distance from the lampost to the woman. and x is the length of the woman's shadow. so here we have similar right triangles. for the first triangle, the two legs are 14' and 18'+x. for the second triangle, the legs are 5' and x'.

set up a proportion for the legs:

14/5 = (18+x)/x

cross multiply:

5(18+x) = 14x

solve for x:

90 +5x=14x

90 = 9x

x =10

her shadow is 10 feet

her friend is taller but the tip of his shadow is at her tip. that could only mean that he is standing closer to the lamp (you can see that from the picture).

from the previous calculations, we came up with x=10. so that means the base of this big triangle is 18+10 =28'. so the length of his shadow plus his distance from the lampost is 28 ft. let's set up a proportion to find out the length of his shadow first. then we can subtract it from 28 to find his distance.

14/6 = 28/x

14x = 168

x = 12

his shadow is 12 ft. subtract that from the base:

28 - 12 = 16 ft. is how far he's standing from the post

Answer 2

This problem solves best with a diagram, but since I can't make one here, let's try something else.

The tip of the 5 foot woman's shadow is unknown distance from her feet, therefore, X.

The woman's feet are 18 feet from the foot of the post.  The light is 14 feet high.

There would be two similar triangles- the legs of the larger one are 14 and (18 + X)

the legs of the smaller one are 5 and X.  Set up a proportion:

14:(18 + X)::5:x

Use product of means equals product of extremes to get:

14X = 5(18 + X)

Solve for X:

X = 10 feet is the length of her shadow

Her friend is 6 feet tall.  Set up a proportion where Y is his distance from the post.

28:14::(28 - Y):6

Solve for Y = 16 feet from the post.

Answer 3

the first answer is:

if x is the length of womans shadow then

x/5 = (x+18)/14

=>x=10 (answer)

if y is the distance between the friend and the woman then

10/5 = (10 +y)/6

=> y= 2

so the shadows length of him is 2+10 =12

the boy is standing 16 ft away from the post. (answer)

Answer 4

OK, make a diagram for this one and its a bunch of triangles so it will be easy

Form a right trangle with the right angle at the base of the light pole.  The heigh of the pole is 14, that is one side of the triangle

Put the 5 ft woman 18 ft away from the pole and then draw two lines.  One that connects the light to the top of the womans head and then continues down

The Other connects the base of the pole (at the right angle) and the womans feet and then continues on until intersecting the first line.  The quantity we want to know is the distance from the womans feet to the intersection of those lines

Lastly draw a third line (make it dotted) from the womans head, parralell with the ground till it intersects the pole.  Notice that this distance is the same as the 18 ft that is known and makes yet another right triangle along with a box.

So the knowns are now

Height of pole - 14

dist to woman from pole - 18

Height of woman - 5

Now you know the dist from top of womans head to top of pole = 14-5 = 9

So now you have a right trangle that has two known sides (9 and 18) so you can determine the angle that the light follows to the woman's head

Tan of this angle  = opposite/adjacent

=18/9

=2

This angle is the same carried from the light, across her head and to the ground, so the 3rd triangle (past the woman, where the two lines intersect) has two knowns....the angle and one side (the womans height)

Its convienient to use the tangent since now the tangent of the angle = opposite/adjacent

tan angle = length of shadow/woman's height

2=length of shadow/5

10 = length of shadow

To do the second one you have a different set of knowns

Height of pole - 14

Height of friend - 6

Length from base of pole to end of shadow - 28 (the shadow above + the original woman's dist to pole)

Draw a new diagram with these values, you don't know the angle or the dist of the person so make it look ok

Since you know the two sides of the triangle (14 and 28) you know everything about the triangle and that the angle must be the same as the previous exercise to make the same total length.  

So tan of the angle (from light top to new persons head) is still = 2

Tan angle = opposite/adjacent



You know the adjacent (poll height - person height) and the angle so you can solve for the dist to pole (opposite)

2=opposite/(14-6)

2=opposite/8

opposite = 16

This makes sense, if the friend is taller they would have to stand closer to the light to get the same total shadow

Just make a diagram, list your knowns and try to make triangles and boxes....puzzles

Answer 5

If you manage to draw the figure properly, you can see that there are two triangles. The smaller triangle is formed when the woman stands perpendicular to the ground and forms a 90 degree angle. Her shadow will make the hypotenuse.

The bigger triangle is formed when the post stands perpendicular to the ground and forms a 90 degree angle. The hypotenuse of the bigger triangle is formed by drawing a straight line from the top of the post through the tip of the woman's height to meet the tip of her shadow. This means that the two triangles share the same angle. Plus, both have a 90 degree angle. So, You can conclude that the two triangles are similar (AA theoem).

Accordingly, you can establish that the height of the post (HP) divided by the height of the woman (HW) is equal to the distance from the bottom of the post (SP) to the tip of the woman's shadow(LSP) divided by the length of the shadow of the woman (LSW).

So, HP/HW=SP/LSW.

The distance from the bottom of the post to the tip of the woman's shadow (SP) is the sum of the distance between the bottom of the post and the woman (18ft) and the length of the shadow of the woman (LSW). -> (18ft+LSW)

Since HP=14ft (assuming the height of the light on top of the post is negligible) and HW=5ft, you can assert that:

14ft/5ft=(18+LSW)/LSW.

=>14LSW=90+5LSW

=>9LSW=90

=>LSW=10

So, the length of the shadow of the woman is 10ft.

For the second situation to be true, the friend has to stand in between the post and the woman. So, what you should do is draw a third line which is perpendicular to the ground in between the two vertical lines and you will have a third triangle.

For this scenario, you need only be concerned with the triangle with the vertical side formed by the friend and the triangle with the vertical side formed by the woman.

If you denote the length of the shadow of the friend to be LSF and his height to be HF, you can establish that (using the same logic for the previous arrangement):

HF/LSF=HW/LSW

=>6ft/LSF=5ft/10ft

=>LSF=12ft

So, the length of the shadow of the friend is 12ft.

It is 28ft (18ft + 10ft) from the bottom of the post to the tip of the shadow of the woman.

And it is 12ft from the point the friend is standing at to the tip of the shadow of the woman.

So, the distance between the friend and the post is the total distance (28ft) minus the distance from the friend to the tip of the woman's shadow (12ft).

Therefore, the friend is standing 16ft far from the post.

Answer 6

1) This is a ratio problem. Set up a triangle to help understand the ratios.

(14-5) / 5 = 18 / x

9 / 5 = 18 / x

Cross multiply

90 = 9x

x= 10

Her shadow is 10 feet.

2) From the previous problem, we know the lightpost casts a shadow of 28 feet.

14 / 6 = 28 / x

168 = 14x

x= 12

The boy casts a 12 foot shadow, so he is 16 feet from the post.

16 feet

Answer 7

Draw a line from the light to the top of the woman's head and continue it to the ground behind her. That line drops 9 feet over a horizontal distance of 18 feet. The question is what horizontal distance is required for it to drop 5 feet. The answer is obviously 10 feet, so that is the length of the woman's shadow.

The ratio of her shadow to his will be 5/6, the length of his shadow will be 10*6/5  = 12 feet and he must therefore be 2 feet close to the light than the woman, so the second answer must be 16 feet