Rate of ascent or descent on a hill

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In summary, we have a conversation about climbing a hill whose shape is described by the equation z=100-0.005x²-0.01y². The coordinates (x,y,z) are measured in meters with the positive x-axis pointing east and the positive y-axis pointing north. We are standing at the point (60,40,966). We are given three questions: (A) If we walk due south, will we start to ascend or descend? At what rate? (B) If we walk due northwest, will we start to ascend or descend? At what rate? (C) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does
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anik18
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30. Suppose you are climbing a hill whose shape is given by the equation z= 100 -0.005x2 - 0.01 y2, where x,y, and z are measured in meters, and you are standing at a point with coordinates (60,40, 966). The positive x-axis points east and the positive y-axis points north.

(A) If you walk due south, will you start to ascend or descend? At what rate?

(B) If you walk due northwest, will you start to ascend or descend? At what rate?

(C) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

Book: Calculus: Concepts and Contexts (3rd) by Stewart

thankyou
 
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Please show some attempted workings. For example, do you understand how to take partial derivative? Do you know what the grad operator does?
 
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(A) Walking due south would result in descending the hill. The rate of descent can be found by taking the partial derivative with respect to y, giving us -0.02y. At the given point, the rate of descent would be -0.02(40) = -0.8 meters per meter traveled south.

(B) Walking due northwest would also result in descending the hill. The rate of descent can be found by taking the partial derivative with respect to both x and y, giving us -0.01x and -0.02y, respectively. At the given point, the rate of descent would be -0.01(60) - 0.02(40) = -1.2 meters per meter traveled northwest.

(C) The slope is largest in the direction of the steepest descent, which can be found by taking the gradient of the function. The gradient is given by ∇z = (-0.01x, -0.02y), and at the given point, the gradient is (-0.6, -0.8). This means that the slope is steepest in the direction of (-0.6, -0.8), which is approximately south-southwest. The rate of ascent in this direction can be found by taking the magnitude of the gradient, which is √((-0.6)2 + (-0.8)2) = 1.0 meters per meter traveled. The angle above the horizontal can be found using trigonometry, giving us an angle of approximately 50.2 degrees.
 

1. What factors affect the rate of ascent or descent on a hill?

The rate of ascent or descent on a hill is affected by the slope or grade of the hill, the weight of the object or person moving, and any external forces such as wind resistance or friction.

2. How does the slope of a hill impact the rate of ascent or descent?

The steeper the slope of a hill, the more difficult it is to ascend or descend, resulting in a slower rate. This is because more energy is required to overcome the gravitational pull of the hill.

3. Does the weight of an object or person affect the rate of ascent or descent on a hill?

Yes, the weight of an object or person has a significant impact on the rate of ascent or descent on a hill. Heavier objects or individuals require more energy to move, resulting in a slower rate of ascent or descent.

4. How do external forces like wind resistance or friction affect the rate of ascent or descent?

External forces such as wind resistance or friction can slow down the rate of ascent or descent on a hill. These forces create resistance, making it more difficult for an object or person to move, thus resulting in a slower rate.

5. Is there an ideal rate of ascent or descent on a hill?

The ideal rate of ascent or descent on a hill depends on various factors, such as the purpose of the movement and the individual's physical abilities. Generally, a steady and controlled rate is recommended to avoid exhaustion or injury.

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