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Values: \(n = 3, 6, 9, 12, 15, 18\) give \(n^3 = 27, 216, 729, 1728, 3375, 5832\) โ all less than 9999. That is 6 values.
Two capacitors $C_1$ and $C_2$ are charged to 120 V and 200 V respectively. When connected together, the potential on each becomes zero. Which relation holds?
For the potential to become zero after connection, the total charge must be zero (charges cancel). Taking signs into account:
$Q_1 - Q_2 = 0 \Rightarrow C_1 \times 120 = C_2 \times 200$
$\frac{C_1}{C_2} = \frac{200}{120} = \frac{5}{3}$
$\Rightarrow 3C_1 = 5C_2$
The capacitors must have been charged with opposite polarities so that when connected, charges cancel. Conservation of charge gives $120C_1 = 200C_2$, yielding $3C_1 = 5C_2$.
The Holliday-Segar method calculates maintenance fluid requirements:
| Weight | Fluid Rate |
|---|---|
| First 10 kg | \(100\,\text{mL/kg/day}\) |
| Next 10 kg (10โ20 kg) | \(50\,\text{mL/kg/day}\) |
| Each kg above 20 kg | \(20\,\text{mL/kg/day}\) |
Calculation for 12 kg child:
\[\text{First 10 kg} = 10 \times 100 = 1000\,\text{mL}\]
\[\text{Next 2 kg} = 2 \times 50 = 100\,\text{mL}\]
\[\text{Total} = 1000 + 100 = 1100\,\text{mL/day}\]
Hourly rate: \(\dfrac{1100}{24} \approx 46\,\text{mL/hr}\)
This formula is essential for pediatric nurses. Remember: it calculates maintenance only โ deficit and ongoing losses are added separately in dehydrated children.
$(19 - 18 - 17 - 16) - (20 - 19 - 18 - 17) =$
Calculate each parenthesis separately:
First: $(19 - 18 - 17 - 16) = 19 - 18 - 17 - 16 = 1 - 17 - 16 = -16 - 16 = -32$
Second: $(20 - 19 - 18 - 17) = 20 - 19 - 18 - 17 = 1 - 18 - 17 = -17 - 17 = -34$
Now subtract: $(-32) - (-34) = -32 + 34 = 2$
Wait, let me recalculate more carefully:
$(19 - 18 - 17 - 16) = 19 - 51 = -32$
$(20 - 19 - 18 - 17) = 20 - 54 = -34$
$(-32) - (-34) = -32 + 34 = 2$
Hmm, 2 is not among the options. Let me check the arithmetic again:
$19 - 18 = 1$, $1 - 17 = -16$, $-16 - 16 = -32$ โ
$20 - 19 = 1$, $1 - 18 = -17$, $-17 - 17 = -34$ โ
$-32 - (-34) = 2$
Since the given answer is D, which corresponds to index 3 (value 1), there may be an error in my reading or the answer key. But mathematically, the answer is 2.
Identify the curve from the family defined by $(x^2-y^2)\,dx + 2xy\,dy = 0$ that passes through the point $(1,1)$.
Rewrite: $\frac{dy}{dx} = \frac{y^2-x^2}{2xy}$. This is a homogeneous ODE. Let $y=vx$:
$\frac{1-v^2}{2v}dx + x\,dv/... $
After solving, the general solution is $x^2 + y^2 = Cy$ (circles with centres on the $y$-axis).
Substituting $(1,1)$: $1+1=C\cdot1 \Rightarrow C=2$, giving $x^2+y^2=2y$, i.e. $x^2+(y-1)^2=1$.
This is a circle centred on the $y$-axis at $(0,1)$.
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