The temperature T in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point (1, 2, 2) is 160°. (a) Find the rate of change of T at (1, 2, 2) in the direction toward the point (4, 1, 3). Incorrect: Your answer is incorrect. (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that

The answers to the questions are as follows;

(a) The rate of change of T at (1, 2, 2) in the direction toward the point (4, 1, 3) is  

(b) The direction of the gradient is in the direction of greatest increase and it is towards the origin.

Explanation:

To solve the question, we note that the shape of the ball is that of a sphere.

Therefore the distance of a point from the center is given by

f(x, y, z)  =

The temperature T in a metal ball is inversely proportional to the distance from the center of the ball

Therefore T ∝ or T =

Where

C = Constant of proportionality

x, y, and z are the x, y and z coordinates values

To find C, we note that at point (1, 2, 2), T = 160 °C.

Therefore 160 °C = = =

Therefore C = 160 × 3 = 480 °C·(Unit length)

We therefore have the general equation as

T =

The vector from points  (1, 2, 2) to point (4, 1, 3) is given by

1·i + 2·j +2·k - (4·i + 1·j +3·k) = -3·i + j -k

From which we find the unit vector given by

u =  

From which we have the gradient equal to

∇T(x, y, z) = -480×(x²+y²+z²) in (x, y, z)

This gives D = ∇T·u

                       = -480×(x²+y²+z²) in (x, y, z)·

That is

(x²+y²+z²) (-3·x + y - z)

From where DTat point (1, 2, 2) is  =  

(b) The direction of greatest increase in temperature is in the direction of the gradient and the direction of the gradient is opposite to the direction of {x, y, z}, which is away from the origin.

Hence the direction of the greatest increase in temperature is towards the origin.

the box does not move because you need to use a force greater than 300 n. you are only using 50 n. it needs to at least be 301 n for the box to move because then it is an unbalanced force and it will move the way that the greater force is pushing it. if you use only 300 n it will be a balanced force and will not move because the force you are exerting on the   box is the same as its force.

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im sorry if i went too in depth, i am going to include a link to a video on khan academy that will explain this.   hope that you like the video and that it explains more in detail to you.

that is the link.

here are some information of the future of geothermal energy:

-the future of geothermal energy can pretty much be summed up with a single word: more

-scientists plan in going deeper than 120 meters

-we are hoping that one day, we will get all of our energy from renewable energy sources such as geothermal energy

hope this !