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Sav [38]
3 years ago
12

C is directly proportional to d such that

Mathematics
1 answer:
defon3 years ago
5 0

Answer:

we have

C = 1.5d

or

d=C/1.5

Step-by-step explanation:

when d is 12

C=1.5×12=18

when c=15

d=15/1.5=10

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Brief review of proportionality relationships:

When two quantities a,b are \textbf{directly} proportional, that means any change in a manifests a \textbf{direct} change (think "in the same direction") in b.

Silly example: "The more I eat, the fatter I get." Here the amount one eats is directly proportional to one's body weight.

This change isn't always one-for-one, so we introduce a constant k to account for any scaling that occurs on either variables behalf. In general, though, we can write a directly proportional relationship as a=kb.

Now, when a,b are \textbf{inversely} proportional, then a change in a manifests a change in b in the \textbf{inverse} (opposite) direction.

Silly example: "The more I eat, the less thin I get."

This time we write the relation as ab=k.

To get back to your problem: To say that the rate of change of y(x) is inversely proportional to \sqrt y is to say that there is some constant k such that

\sqrt y\dfrac{\mathrm dy}{\mathrm dx}=k

This is a separable ODE:

y^{1/2}\,\mathrm dy=k\,\mathrm dx
\displaystyle\int y^{1/2}\,\mathrm dy=\int k\,\mathrm dx
\dfrac23y^{3/2}+C_y=kx+C_x
\dfrac23y^{3/2}=kx+C
y^{3/2}=\dfrac{3k}2x+C
y=\left(\dfrac{3k}2x+C\right)^{2/3}
8 0
3 years ago
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