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jek_recluse [69]
3 years ago
15

Suppose you are climbing a hill whose shape is given by the equation z = 900 − 0.005x2 − 0.01y2, where x, y, and z are measured

in meters, and you are standing at a point with coordinates (120, 80, 764). 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? ascend descend Correct: Your answer is correct.
Mathematics
1 answer:
Kazeer [188]3 years ago
8 0

Answer:

Ascend

Step-by-step explanation:

In order to solve this problem, we are going to use some principles of vector calculation. The concepts we are going to use are Partial derivatives, gradient vector, velocity vector, direction vector, and directional derivative.

The gradient vector is a vector that describes how is the 'slope' in the space of a multivariable function at a specified point; it is built as a vector of partial derivatives. The vector velocity is a vector that describes the direction and speed of the movement of a body, if we make the velocity a unitary vector (a vector whose norm is 1), we obtain the direction vector (because we are not considering the real norm of the vector, just direction). Finally, the directional derivative is a quantity (a scalar) that describes the slope that we get on a function if we make a displacement from a particular point in a specific direction.  

The problem we have here is a problem where we want to know how will be the slope of the hill if we stand in the point (120, 80, 764) and walk due south if the hill has a shape given by z=f(x,y). As you see, we have to find the directional derivative of z=f(x,y) at a specific point (120, 80, 764) in a given displacement direction; this directional derivative will give us the slope we need. The displacement direction 'u' is (0,-1): That is because 'y' axis points north and our displacement won't be to the east either west (zero for x component), just to south, which is the opposite direction of that which the y-axis is pointing (-1 for y component). Remember that the direction vector must be a unitary vector as u=(0,-1) is.

Let's find the gradient vector:

z=900-0.005x^2-0.01y^2\\\frac{\partial z}{\partial x}=-0.005*2*x=-0.01x\\\frac{\partial z}{\partial y}=-0.01*2*y=-0.02y\\ \nabla (z)=(-0.01x,-0.02y)

Evaluate the gradient vector at (120,80) (764 is z=f(120,80); you may confirm)

\nabla (z(120,80))=(-0.01*120,-0.02*80)=(-1.2,-1.6)

Finally, find the directional derivative; if you don't remember, it can be found as a dot product of the gradient vector and the direction vector):

D_{u,P_0}= \nabla (z)_{P_0}\cdot u\\D_{u,P_0}= (-1.2,-1.6)\cdot (0,-1)=1.6

As you see, the slope we find is positive, which means that we are ascending at that displacement direction.

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I believe the ansr would be (-6,-4) I say this becuase if it's positive on the y axis the X axis will be negative and if it's being reflected the coordinates will be backwards. If this answer is wrong I'm truly sorry, but I hope this helps!
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3 years ago
Mary and Roberto go to the movies. They buy a large popcorn for $8.50, and two medium sodas for $4.25. How much did they spend a
alukav5142 [94]

Answer:

$19.15

Step-by-step explanation:

Equation 1: 8.50+4.25+4.25= 17.00  (price of the popcorn and drinks)

Equation 2: 17.00/7.89= 2.15                       (how to find how much the tax is)

Solve: 17.00+2.15=19.15                     (add the tax to how much the popcorn and drinks were)

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What is 97.356 in base ten numerals
nadya68 [22]
97.356

<span>ninety seven and three hundred fifty six <span>thousandths</span></span>

4 0
3 years ago
I have 100 items of product in stock. The probability mass function for the product's demand D is P(D=90)=P(D=100)=P(D=110)=1/3.
masya89 [10]

Answer:

The probability mass function for the items sold is

P_X(k) = \left \{ {\frac{1}{3} \, \, \, {k=90} \atop \, \frac{2}{3} \, \, \, {k=100}} \right.

The mean is 96.667

The variance is 22.222

b) The probability mass function for the unfilled demand due to lack of stock is

P_Y(k) = \left \{ {\frac{2}{3} \, \, \, {k=0} \atop \, \frac{1}{3} \, \, \, {k=10}} \right.

The mean is 3.333

The variance is 33.333

Step-by-step explanation:

If the demand is higher than 100, then you will sell 100 items only. Thus, there is a probability of 1/3+1/3 = 2/3 that you will sell 100 items, while there is a probability of 1/3 that you will sell 90.

The probability mass function for the items sold is

P_X(k) = \left \{ {\frac{1}{3} \, \, \, {k=90} \atop \, \frac{2}{3} \, \, \, {k=100}} \right.

The mean is 1/3 * 90 + 2/3 * 100 = 290/3 = 96.667

The variance is V(X) = E(X²)-E(X)² = (1/3*90² + 2/3*100²) - (290/3)² = 200/9 = 22.222

b) If order to be unfilled demand, you need to have a demand of 110, which happens with probability 1/3. In that case, the value of the variable, lets call it Y, that counts the amount of unfilled demand due to lack of stock is 110-100 = 10. In any other case, the value of Y is 0, which would happen with probability 1-1/3 = 2/3. Thus

P_Y(k) = \left \{ {\frac{2}{3} \, \, \, {k=0} \atop \, \frac{1}{3} \, \, \, {k=10}} \right.

The mean is 2/3 * 0 + 1/3 * 10 = 10/3 = 3.333

The variance is 2/3*0² + 1/3*10² = 100/3 = 33.333

4 0
3 years ago
18
Roman55 [17]

Answer:

  2.53×10^8 liters

Step-by-step explanation:

  (8.65×10^3 L/da)(0.292×10^5 da) ≈ 2.53×10^8 L

3 0
3 years ago
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