A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75m/s. a.)at what rate is the lenght of her shadow increasing? b.)at what speed is the tip of her shadow moving?

A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of 0.75m/s. a.)at what rate is the lenght of her shadow increasing? b.)at what speed is the tip of her shadow moving?

Question
Similarity
asked 2021-02-27
A spotlight at a building corridor is fastened to a wall 8m above the floor. A lady 1.75m tall moves away from the wall at a speed of \(\displaystyle{0.75}\frac{{m}}{{s}}\).
a.)at what rate is the lenght of her shadow increasing?
b.)at what speed is the tip of her shadow moving?

Answers (1)

2021-02-28
Step 1
Please have a look at the picture below to understand what's going on:
image
For the sake of clarity,
s = length of the shadow = AD
and x = distance of the lady from the wall = BD
\(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={0.75}\frac{{m}}{{s}}\)
Step 2
Triangle ADE is similar to triangle ABC (AAA criterion of similarity)
Hence, \(\displaystyle{D}\frac{{E}}{{B}}{C}={A}\frac{{D}}{{A}}{B}\) (Corresponding sides of similar triangles are proportional)
Hence, \(\displaystyle\frac{{1.75}}{{8}}=\frac{{s}}{{{s}+{x}}}\)
Hence, \(\displaystyle{1.75}{\left({s}+{x}\right)}={8}{s}\)
Or, \(\displaystyle{1.75}{x}={\left({8}-{1.75}\right)}{s}={6.25}{s}\)
Step 3
Part (a)
Differentiate both sides w.r.t time t to get:
\(\displaystyle\frac{{{1.75}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}=\frac{{{6.25}{d}{s}}}{{{\left.{d}{t}\right.}}}\)
Hence, the rate at which length of her shadow is increasing = \(\displaystyle\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}=\frac{{{\left(\frac{{1.75}}{{6.25}}\right)}{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}={\left(\frac{{1.75}}{{6.25}}\right)}{x}{0.75}={0.21}\frac{{m}}{{s}}\)
Step 4
Part (b)
the rate at which the tip of her shadow is moving = rate t which she is moving + rate at which the length of the shadow is increasing = \(\displaystyle\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}+\frac{{{d}{s}}}{{{\left.{d}{t}\right.}}}={0.75}+{0.21}={0.96}\frac{{m}}{{s}}\)
Step 5
Final answers:
Part (a) \(\displaystyle{0.21}\frac{{m}}{{s}}\)
Part (b) \(\displaystyle{0.96}\frac{{m}}{{s}}\)
0

Relevant Questions

asked 2021-02-23
A 0.30 kg ladle sliding on a horizontal frictionless surface isattached to one end of a horizontal spring (k = 500 N/m) whoseother end is fixed. The ladle has a kinetic energy of 10 J as itpasses through its equilibrium position (the point at which thespring force is zero).
(a) At what rate is the spring doing work on the ladle as the ladlepasses through its equilibrium position?
(b) At what rate is the spring doing work on the ladle when thespring is compressed 0.10 m and the ladle is moving away from theequilibrium position?
asked 2021-04-13
As depicted in the applet, Albertine finds herself in a very odd contraption. She sits in a reclining chair, in front of a large, compressed spring. The spring is compressed 5.00 m from its equilibrium position, and a glass sits 19.8m from her outstretched foot.
a)Assuming that Albertine's mass is 60.0kg , what is \(\displaystyle\mu_{{k}}\), the coefficient of kinetic friction between the chair and the waxed floor? Use \(\displaystyle{g}={9.80}\frac{{m}}{{s}^{{2}}}\) for the magnitude of the acceleration due to gravity. Assume that the value of k found in Part A has three significant figures. Note that if you did not assume that k has three significant figures, it would be impossible to get three significant figures for \(\displaystyle\mu_{{k}}\), since the length scale along the bottom of the applet does not allow you to measure distances to that accuracy with different values of k.
asked 2021-04-19
For the cellar of a new house a hole is dug in the ground, withvertical sides going down 2.40m. A concrete foundation wallis built all the way across the 9.6m width of the excavation. This foundation wall is 0.183m away from the front of the cellarhole. During a rainstorm, drainage from the streetfills up the space in front of the concrete wall, but not thecellar behind the wall. The water does not soak into the clays oil. Find the force the water causes on the foundation wall. For comparison, the weight of the water is given by:
\(\displaystyle{2.40}{m}\cdot{9.60}{m}\cdot{0.183}{m}\cdot{1000}{k}\frac{{g}}{{m}^{{3}}}\cdot{9.8}\frac{{m}}{{s}^{{2}}}={41.3}{k}{N}\)
asked 2021-05-16
Look Out! A snowball rolls off a barn roof that slopes downward at an angle of 40 degrees . The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m/s as it rolls off the roof. Ignore air resistance.
A man 1.9 m tall is standing 4.0 m from the edge of the barn. Will he be hit by the snowball?
asked 2021-02-18
In an industrial cooling process, water is circulated through a system. If the water is pumped with a speed of 0.45 m/s under a pressure of 400 torr from the first floor through a 6.0-cm diameter pipe, what will be the pressure on the next floor 4.0 m above in a pipe with a diameter of 2.0 cm?
asked 2021-04-16
A child is playing on the floor of a recreational vehicle (RV) asit moves along the highway at a constant velocity. He has atoy cannon, which shoots a marble at a fixed angle and speed withrespect to the floor. The cannon can be aimed toward thefront or the rear of the RV. Is the range toward the frontthe same as, less than, or greater than the range toward the rear?Answer this question (a) from the child's point of view and (b)from the point of view of an observer standing still on the ground.Justify your answers.
asked 2021-03-12
A 75.0-kg man steps off a platform 3.10 m above the ground. Hekeeps his legs straight as he falls, but at the moment his feettouch the ground his knees begin to bend, and, treated as aparticle, he moves an additional 0.60 m before coming torest.
a) what is the speed at the instant his feet touch theground?
b) treating him as a particle, what is his acceleration(magnitude and direction) as he slows down, if the acceleration isassumed to be constant?
c) draw his free-body diagram (see section 4.6). in termsof forces on the diagram, what is the net force on him? usenewton's laws and the results of part (b) to calculate the averageforce his feet exert on the ground while he slows down. expressthis force in newtons and also as a multiple of his weight.
asked 2021-05-13
A movie stuntman (mass 80.0kg) stands on a window ledge 5.0 mabove the floor. Grabbing a rope attached to a chandelier, heswings down to grapple with the movie's villian (mass 70.0 kg), whois standing directly under the chandelier.(assume that thestuntman's center of mass moves downward 5.0 m. He releasesthe rope just as he reaches the villian).
a) with what speed do the entwined foes start to slide acrossthe floor?
b) if the coefficient of kinetic friction of their bodies withthe floor is 0.250, how far do they slide?
asked 2021-04-06
A race car enters a flat 200 m radius curve at a speed of 20 m/swhile increasing its speed at a constant 2 m/s2. If the coefficient of static friction is .700, what will the speed of thecar be when the car beings to slide?
A) 24.3 m/s
B) 31.5 m/s
C) 37.1 m/s
D) 36.2 m/s
E) 28.7 m/s
asked 2021-02-19
An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of \(\displaystyle{1950}\ {N}\cdot{m}\) to the propeller, which starts from rest.
a) What is the angular acceleration of the propeller? Model the propeller as a slender rod.
b) What is the propeller's angular speed after making 5.00 revolutions?
c) How much work is done by the engine during the first 5.00 revolutions?
e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?
...