1) Suppose a plasma bubble with a radius of r (mm) and mas of m (mg) is predicte

Question

1) Suppose a plasma bubble  with a radius of r (mm) and mas of m (mg) is predicted to have a temperature of T degrees according to the formula : root (T)=m^(5/4)r^(3/2)  

note: a  constant that corrects units has been corrected to 1.

If the mass and radius are given as m=5 and r=6, and the rates of change of the temperature and the radius are given as 2deg/sec and 1mm/s what is the rate of change of the mass?

2) A particle moves in the plane along the curve y=sin(8x).  At what rate is its distance from the origin increasing at the moment its x-coordinate is 2 and its x-coordinate is increasing at a speed of 8m/s?

Explanation / Answer

1.) since root T = m^(5/4) r^(3/2)

if m = 5, r = 6 then T = 12074.77 degree

then dT/ 2root T = (5/4) m^(1/4) * r^(3/2) dm + (3/2) r^(1/2)*m^(5/4) dr

putting the values

2 / 2x109.89 = (5/4) (1.5) (14.7) (dm) + (3/2) (2.45) (7.48)(1)

so dm = -0.99 means 1 mg/sec

2.) dy/dx = 8 cos(8x)

so dy/8 = 8 cos(16)

so dy = 61.52 m/s

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