Suppose the radiation level (in rads) in some region of space is given by R = f(

Question

Suppose the radiation level (in rads) in some region of space is given by R = f(x, y, z) = x^2 + yz, where (x, y, z) are Cartesian coordinates measured in kilometers. A space ship is sitting at location (1, 6, 3). a) At what rate will the radiation level begin to change (in rads/km) if the ship moves in the direction of the vector (2, -1, 1)? In other words, what is D_u f(1, 6, 3), where u is in the direction of (2, -1, 1)? b) In which vector direction should the ship move in order to begin decreasing the radiation level as rapidly as possible? In other words, find the direction of maximum decrease of f at (1, 6, 3).

Explanation / Answer

a) Given, f(x, y, z)=x²+yz

So, f(x, y, z)=<2x, z, y>

then at point (1, 6, 3) we have, f(1, 6, 3) =<2, 3, 6>

Now, the unit vector in the direction of <2, -1, 1> is u=1/(4+1+1)<2, -1, 1>=<2/6, -1/6, 1/6>

Then,

Duf(1, 6, 3)=f(1, 6, 3).u=<2, 3, 6>.<2/6, -1/6, 1/6>=<4/6, -3/6, 6/6>

b) The direction of maximum increase in radiation at the point (1, 6, 3) is given f(1, 6, 3), so the direction of rapidly decrease is -f(1, 6, 3)=-<2, 3, 6>=< -2, -3, -6>

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.