Respuesta :
Answer:
A ( t ) = -4t + 51
B ( t ) = -2t + 29
t < 11 hours ... [ 0 , 11 ]
Step-by-step explanation:
Given:-
- The height of snowman A, Ao = 51 in
- The height of snowman B, Bo = 29 in
Solution:-
- The day Mason made two snowmans ( A and B ) with their respective heights ( A(t) and B(t) ) will be considered as the initial value of the following ordinary differential equation.
- To construct two first order Linear ODEs we will consider the rate of change in heights of each snowman from the following day.
- The rate of change of snowman A's height ( A ) is:
[tex]\frac{d h_a}{dt} = -4[/tex]
- The rate of change of snowman B's height ( B ) is:
[tex]\frac{d h_b}{dt} = -2[/tex]
Where,
t: The time in hours from the start of melting process.
- We will separate the variables and integrate both of the ODEs as follows:
[tex]\int {} \, dA= -4 * \int {} \, dt + c\\\\A ( t ) = -4t + c[/tex]
[tex]\int {} \, dB= -2 * \int {} \, dt + c\\\\B ( t ) = -2t + c[/tex]
- Evaluate the constant of integration ( c ) for each solution to ODE using the initial values given: A ( 0 ) = Ao = 51 in and B ( 0 ) = Bo = 29 in:
[tex]A ( 0 ) = -4(0) + c = 51\\\\c = 51[/tex]
[tex]B ( 0 ) = -2(0) + c = 29\\\\c = 29[/tex]
- The solution to the differential equations are as follows:
A ( t ) = -4t + 51
B ( t ) = -2t + 29
- To determine the time domain over which the snowman A height A ( t ) is greater than snowman B height B ( t ). We will set up an inequality as follows:
A ( t ) > B ( t )
-4t + 51 > -2t + 29
2t < 22
t < 11 hours
- The time domain over which snowman A' height is greater than snowman B' height is given by the following notation:
Answer: [ 0 , 11 ]
