Question

10) Un balón esférico es inflado con gas a razón de 6 (cms) por minuto Cuan rápido el radio del balón crece en el instante en que el radio alcanza los 5 cms? Pasol Identificar datos conocidos y desconocidos Paso 2 Escribir una ecuacion que relacione las variables conocidas y desconocidas Paso3 Derivar implícitamente Paso 4 Sustituire interpretar la respuesta

A spherical balloon is inflated with gas at a rate of 6 (???) 3 per minute. How fast does the radius of the balloon grow the instant the radius reaches 5 cm?

Step1 Identify known and unknown data

Step 2

Write an equation that relates the known and unknown variables

Step 3

Implicitly derive

Step 4 Substitute and interpret the answer

Answer 1

Step 1: known Data:

Rate at which Volume of balloon is increasing = dV/dt = +6 cm^3/min

Radius of balloon at required point = R = 5 cm

Unknown Data:

Rate at which radius of balloon is increasing = dr/dt = ?

Step 2: Relation between Volume and Radius of sphere is given by

V = (4/3)*pi*R^3

Step 3: Using Implicit differentiation find rate of change in Volume and Radius

dV/dt = d[4*pi*R^3/3]/dt

dV/dt = (4*pi*3*R^2/3)*dR/dt

dV/dt = 4*pi*R^2*(dR/dt)

dR/dt = (1/(4*pi*R^2))*dV/dt

Step 4: Using known values:

R = 5 cm & dV/dt = 6 cm^3/min

Which gives

dR/dt = (1/(4*pi*5^2))*6

dR/dt = 0.019 cm/min = rate at which radius is increasing

Let me know if you've any query.