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Ncert Solutions Maths class 12th Maths Class 12th Application of Derivatives

8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm

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Vishal Baghel | Contributor-Level 10

4 months ago

Let ‘r’ cm be the radius of volume V. measured Then,

$V = \frac{4}{3} π r^{3}$

Now, rate at which balloon is being inflated = 900 ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d v}{d r} = 900$ ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d t} (\frac{4}{3} π r^{3})$ = 900 ${cm}^{\frac{3}{5}}$

$\Rightarrow \frac{d}{d r} (\frac{4}{3} \times r^{3}) \frac{d r}{d t} = 900$

$\frac{4}{3}$ $π$ × 3 × r² $\frac{d r}{d r}$ = 900.

$\Rightarrow \frac{d r}{d t} = \frac{900}{4 π r^{2}} .$

When r = 15cm,

${\frac{d s}{d t} |}_{r = 15} = \frac{900}{4 π \times {(15)}^{2}}$ = $\frac{900}{900 π} = \frac{1}{π}$ cm/s.

Q Radius of balloon increases by $\frac{1}{π}$ per second.

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If the normal to the curve y(x) = ∫(from 2 to x) (2t² - 15t + 10)dt at a point (a, b) is parallel to the line x + 3y = -5, a > 1, then the value of |a + 6b| is equal to..........

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y (x) = ∫? (2t² - 15t + 10)dt
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2a² - 15a + 10 = -3
2a² - 15a + 13 = 0 (The provided solution has 2a²-15a+7=0, suggesting a different problem or a typo)
Following the image: 2a² - 15a + 7 = 0
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b = ∫? (2t² - 15t + 10)dt = [2t³/3 - 15t²/2 + 10t] from 0 to 7.
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Let F₁(A,B,C) = $(A \land \sim B) \lor$ $[\sim C \land (A \lor B)] \lor \sim A a n d F_{2} (A, B) = (A \lor B) \lor (B \to \sim A)$ be two logical expressions. Then:

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By truth table

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From option let it be isosceles where AB = AC then

$x = \sqrt[]{r^{2} - {(h - r)}^{2}}$

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So $A B = \sqrt[]{h^{2} + x^{2}} = \sqrt[]{\frac{9 r^{2}}{4} + \frac{3 r^{2}}{4}} = \sqrt[]{3} r$ .

Hence $Δ$ be equilateral having each side of length $\sqrt[]{3} r .$

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8. A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm

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