If #V=1.5*t+0.0080*t^2# has the units millions of cubic feet per month, how would you rewrite the equation if you wanted the result in cubic feet per second?

Question

Julia Aguilar · Accepted Answer

#V=3.888*t + 5,374.8*t^2#

The volume, #V#, is given as #10^6# cubic feet. Converting this to just cubic feet requires us to divide #V# by a factor of #10^6#.

Converting months to seconds requires several multiplying factors

#(30 cancel(" days"))/(1 " month")xx(24 cancel(" hours"))/(1 cancel(" day"))xx(60 cancel(" min"))/(1 cancel(" hour"))xx(60 " sec")/(1 cancel(" min"))# #=2.592xx10^6 "sec"/"month"#

Multiplying the volume, #V#, and time, #t# by these factors gives

#V/10^6=2.592xx10^6*1.5*t+(2.592xx10^6)^2*0.008*t^2#

#V=10^-6(2.592xx10^6*1.5*t+(2.592xx10^6)^2*0.008*t^2)# #V=2.592*1.5*t+10^-6*(2.592xx10^6)^2*0.008*t^2# #V=2.592*1.5*t+(2.592)^2*0.008xx10^6t^2# #V=3.888*t + 5,374.8*t^2#

Charles Aeschliman · Answer

undefined To rewrite the equation in cubic feet per second, you need to convert the time units from months to seconds. Since 1 month is equal to approximately 2,628,000 seconds, you can substitute this conversion factor into the equation:

$$ V = 1.5t + 0.0080t^2 $$

Substitute $ t $ with $ t_{\text{months}} $ converted to seconds:

$$ V = 1.5 \left(\frac{t_{\text{months}}}{2628000}\right) + 0.0080 \left(\frac{t_{\text{months}}}{2628000}\right)^2 $$

Simplify:

$$ V = \frac{1.5t_{\text{months}}}{2628000} + \frac{0.0080t_{\text{months}}^2}{(2628000)^2} $$

$$ V = \frac{1.5}{2628000}t_{\text{months}} + \frac{0.0080}{(2628000)^2}t_{\text{months}}^2 $$

Since $ t_{\text{months}} $ is now in seconds, the units of $ V $ will be cubic feet per second.