A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. [Figure: lamp-post AB vertical, girl CD vertical standing away from the post; ray from lamp top A grazes girl's head C and meets ground at E; DE is the shadow.]
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Step-by-Step Solution
Step 1: Convert girl's height to meters
The girl's height is given in centimeters, but all other lengths are in meters. To maintain consistency, we convert the girl's height to meters. Since 1 m=100 cm, we divide 90 cm by 100 to get 0.9 m.
Step 2: Calculate distance walked by the girl
The girl is walking away from the lamp-post at a constant speed. We need to find the distance she has covered in the given time. We use the formula: distance equals speed multiplied by time. This distance is represented as BD in the diagram.
Step 3: Calculate the value of BD
Multiplying the speed 1.2 m/s by the time 4 s, we find that the distance travelled by the girl from the base of the lamp-post is 4.8 meters.
Step 4: Identify similar triangles
We have two right-angled triangles: △ABE (formed by the lamp-post, the ground, and the light ray) and △CDE (formed by the girl, the ground, and her shadow). Both the lamp-post and the girl are perpendicular to the ground, so ∠B=∠D=90∘. Also, ∠E is common to both triangles. Therefore, by the AA similarity criterion, △ABE is similar to △CDE.
Step 5: Set up proportionality of sides
Since △ABE∼△CDE, the ratio of their corresponding sides must be equal. Here, AB is the height of the lamp-post, CD is the height of the girl, BE is the total distance from the lamp-post to the end of the shadow, and DE is the length of the shadow.
Step 6: Substitute known values
We substitute the given values into the proportion. The height of the lamp AB=3.6 m, the height of the girl CD=0.9 m. The distance BE is the sum of BD (distance girl walked) and DE (length of shadow), so BE=4.8+DE. Let DE be x for calculation ease.
Step 7: Simplify and solve for DE
Simplify the ratio 03.6.9 to 4. Then, multiply both sides by DE to eliminate the denominator. This yields a linear equation 4×DE=4.8+DE. Subtract DE from both sides (4×DE−DE=3×DE=4.8). Finally, divide 4.8 by 3 to solve for DE.
Step 8: Calculate the length of the shadow
Performing the division, 4.8÷3 equals 1.6. Thus, the length of the girl's shadow after 4 seconds is 1.6 meters.