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allsm [11]
2 years ago
15

This is a Differential Equations problem, I only need help on part 1 from question five. I need steps as well, thank you.

Mathematics
1 answer:
jolli1 [7]2 years ago
6 0

Answer:

a) y(t) =  \sqrt{2t + 1 }

b) 1.5 hours after thickness will be 2 inches.

Step-by-step explanation:

\frac{dy}{dt}  =  \frac{1}{y}   \\   \\ ydy = dt  \\ \\ integrating \: both \: sides \\   \\  \int y \: dy =  \int 1 \: dt \\  \\  \frac{ {y}^{2} }{2}  = t + c \\  \\  {y}^{2}  = 2t + 2c \\  \\ y (t)=  \sqrt{2t + 2c} ...(1) \\  \\ a)  \:  \: \: plug \: t = 0 \: in \: (1) \\  \\  y (0)=  \sqrt{2(0) +2 c} \\  \\ y (0)=  \sqrt{0 +2 c} \\  \\ 1 =  \sqrt{2c}  \:  \: ( \because \: y (0)=1) \\  \\  2c = 1   \:  \implies \: c =  \frac{1}{2} \\  \\ plug \: c = \frac{1}{2} \: in \: (1) \\  \\ y(t) =  \sqrt{2t + 2 \times  \frac{1}{2} } \\  \\  \huge \:  \red {y(t) =  \sqrt{2t + 1 } } \\  \\b) \:  \:  plug \: y(t) = 2 \: in \: above \: equation \\  \\ 2 =  \sqrt{2t + 1}  \\  \\ 4 = 2t + 1 \: \\  (squaring \: both \: sides) \\  \\ 4 - 1 = 2t \\  \\ 2t = 3 \\  \\ t =  \frac{3}{2}  \\  \\ t = 1.5 \: hours \\

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