Answers to any of these questions would be of great help! (:
1 a) At noon, ship A is 150km west of ship B. Ship A is sailing south at 50km/h and ship B is sailing north at 30km/h. How fast is the distance between the ships changing at 4:00pm?
b) A spotlight on the ground is shining on a wall 10m away. If a woman 2m tall walks from the spotlight toward the building at a speed of 0.7m/s, how fast is the length of her shadow on the building decreasing when she is 5m from the building?
c) Brain weight B as a function of body weight W in fish has been modeled by the power function B=0.007W^(2/3), where B and W are measured in grams. A model for body weight as a function of body length L (measured in cm) is W=0.12L^2.53. If, over 10 million years, the average length of a certain species of fish evolved from 15cm to 30cm at a constant rate, how fast was the species' brain growing when the average length was 21cm? Round your answer to the nearest hundredth. (in nanograms per year)
Thank you in advance! Help is much appreciated!
a) Draw a diagram. At time t, A is 50t km south of the line which joined the ships at noon. B is 30t km north of that line, which is 150km long.
So, the distance d is such that
d^2 = 150^2 + (80t)^2
at 4:00,
d^2 = 150^2 + 320^2 = 124900
d = 353.4
2d dd/dt = 12800t
dd/dt = 12800t/2d
At 4:00, t=4
dd/dt = 12800*4/706.8 = 72.44
This makes sense, since the farther apart the two ships get, the closer their separation speed gets to just plain 50+30 = 80 km/hr
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b) Using similar triangles, when the woman is x meters from the light, the height h on the wall satisfies:
2/x = h/10
xh = 20
h dx/dt + x dh/dt = 0
Now, when she's 5m from the building, she's 5m from the light, so x=5 and
2/5 = h/10
h = 4
4*.75 + 5 dh/dt = 0
dh/dt = -3/5 = -0.6 m/s
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c)*whew*
The length grows at a rate
dL/dt = 15cm/10^6 yrs
W = .12L2.53
B = .007W2/3
= .007(.12L2.53)2/3
= 0.00168L1.69
So, when L = 21
dB/dt = .00168*1.69L.69dL/dt
= .00284*21.69*15/10^6 = 0.000000348129
= 348.13 ng/yr
thank you steve!
For question a), how did you get 12800? Where'd that come from?
Sure! I'd be happy to help you with these questions.
1a) In order to find the rate at which the distance between Ship A and Ship B is changing, we can use the concept of related rates. Let's denote the distance between the ships as D(t), where t represents time. Ship A is sailing south at a speed of 50 km/h, so its position can be described as x(t) = -50t (since west is considered negative). Similarly, Ship B is sailing north at a speed of 30 km/h, so its position is y(t) = 30t. The distance between the ships can be found using the distance formula: D(t) = sqrt((x(t)-y(t))^2). Now we need to find dD/dt, the rate at which the distance is changing with respect to time. We can differentiate D(t) with respect to t and substitute the given values to find the rate of change at 4:00pm.
b) To determine how fast the length of the woman's shadow on the building is decreasing, we can use similar concepts of related rates. Let's denote the length of the shadow as S(t), where t represents time. The distance from the woman to the building can be represented as x(t) = 10 - 0.7t (since she is moving towards the building). We need to find dS/dt, the rate at which the length of the shadow is changing with respect to time. We can differentiate S(t) with respect to t and substitute the given values to find the rate of change when the woman is 5m from the building.
c) To calculate the rate at which the fish species' brain is growing when the average length is 21cm, we need to find the derivative of brain weight (B) with respect to time. We are given the relationship between brain weight and body weight (B = 0.007W^(2/3)) and the relationship between body weight and body length (W = 0.12L^2.53). Since the length is changing with time, we need to find dL/dt to substitute into the equations. We can then calculate dB/dt, the rate at which brain weight is changing with respect to time, and convert it to nanograms per year by appropriate conversions.
To get the actual numerical answers, you'll need to plug in the specific values at the given time or length and solve the equations.