Study questions platform-wide or filter by specific tests with correct answers revealed.
Given $(2 + \sin x)\dfrac{dy}{dx} + (y+1)\cos x = 0$ with $y(0) = 1$, find the value of $y\!\left(\dfrac{\pi}{2}\right)$.
Separate variables: $\frac{dy}{y+1} = -\frac{\cos x}{2+\sin x}dx$
Integrate: $\ln|y+1| = -\ln|2+\sin x| + C$
So $(y+1)(2+\sin x) = K$
At $x=0, y=1$: $(1+1)(2+0) = K \Rightarrow K = 4$
Thus $y+1 = \frac{4}{2+\sin x}$, so $y = \frac{4}{2+\sin x} - 1$
At $x = \frac{\pi}{2}$: $y = \frac{4}{2+1} - 1 = \frac{4}{3} - 1 = \mathbf{\frac{1}{3}}$
Runner A completed a distance of $\dfrac{4}{5}$ kilometer, while Runner B ran 800 meters. Compare the distances covered by each runner.
Quantity A: The distance Runner A ran
Quantity B: The distance Runner B ran
Convert both distances to the same unit.
Runner A ran $\dfrac{4}{5}$ km $= 0.8$ km $= 800$ meters.
Runner B ran $800$ meters.
Both runners covered exactly the same distance: 800 meters.
Therefore the two quantities are equal.
Sign in to join the conversation and share your thoughts.
Log In to Comment