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!
sorry accidentally reposted!!
Sure! I can help you with these questions. Let's go through each of them step by step.
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. We want to find how fast the distance between the ships is changing at 4:00pm.
To solve this problem, we can use the concept of relative velocity. We need to find the rate of change of the distance between the ships with respect to time. Let's assume the distance between them at time t is D(t).
The rate of change of the distance between the ships, dD/dt, is given by:
dD/dt = speed of ship A + speed of ship B
Since ship A is sailing south and ship B is sailing north, their speeds have opposite signs. So, the equation becomes:
dD/dt = 50 km/h + (-30 km/h) = 20 km/h
Therefore, the distance between the ships is changing at a rate of 20 km/h.
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, we want to find how fast the length of her shadow on the building is decreasing when she is 5m from the building.
To solve this problem, we can use similar triangles and the concept of related rates. Let's assume the length of the shadow at time t is S(t).
We know that the woman's height, H, and the distance from the spotlight to the wall, D, are constant. So, we need to find dS/dt, the rate of change of the length of the shadow with respect to time.
Using the concept of similar triangles, we have:
S(t)/H = (S(t) + x)/(H + 2)
Simplifying the equation, we get:
S(t)(H + 2) = H(S(t) + x)
Now, differentiate both sides with respect to time:
d(S(t)(H+2))/dt = d(H(S(t)+x))/dt
Using the product rule, we have:
dS/dt * (H+2) + S * d(H+2)/dt = H * dS/dt + S * dx/dt
We are given that dx/dt = -0.7 m/s (negative because the woman is moving towards the building) and we want to find dS/dt when S = 5m. We also know that H = 2m.
Substituting the values, we have:
dS/dt * (2 + 2) + 5 * 0 = 2 * dS/dt + 5 * (-0.7)
Simplifying the equation, we get:
4dS/dt = -3.5
Therefore, the length of the shadow on the building is decreasing at a rate of -0.875 m/s when the woman 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), and body weight W as a function of body length L is given by W = 0.12L^2.53. We want to find how fast the species' brain was growing in nanograms per year when the average length was 21cm.
To solve this problem, we can use the chain rule of differentiation. Let's assume the average length of the fish at time t is L(t), and the brain weight is B(t).
We want to find dB/dt, the rate of change of brain weight with respect to time.
First, find dW/dL, the derivative of body weight with respect to body length. Using the power rule of differentiation, we have:
dW/dL = 0.12 * 2.53L^(2.53-1) = 0.12 * 2.53L^1.53
Next, find dB/dW, the derivative of brain weight with respect to body weight. Using the power rule of differentiation, we have:
dB/dW = 0.007 * (2/3)W^(-1/3)
Now, we can find dB/dt using the chain rule:
dB/dt = (dB/dW) * (dW/dL) * (dL/dt)
Substituting the values we have:
dB/dt = (0.007 * (2/3)(W^(-1/3))) * (0.12 * 2.53L^1.53) * (dL/dt)
We are given that the length of the fish is changing at a constant rate, so dL/dt is constant. We also know that L = 21cm, so substitute these values.
Rounding your answer to the nearest hundredth, you can evaluate the expression to find the rate of growth of the species' brain weight in nanograms per year.
I hope this helps! Let me know if you have any further questions.