Explanation

(a)

(i) Using volume of reservoir = volume of hemisphere + volume of cone.

\(\implies 333\frac{1}{3}\pi = \frac{2}{3}\pi r^{3} + \frac{1}{3}\pi r^{2} h\)

\(\implies \frac{1000}{3}\pi = \frac{1}{3}\pi (2r^{3} + r^{2}h)\)

Let the radius of the hemisphere be x m;

Then \(\frac{1000}{3} = \frac{1}{3}(2x^{3} + x^{2}(6x))\)

\(1000 = 2x^{3} + 6x^{3}\)

\(1000 = 8x^{3} \implies x^{3} = \frac{1000}{8} = 125\)

\(x = \sqrt[3]{125} = 5 m\)

Hence, the volume of the hemisphere = \(\frac{2}{3}\pi r^{3}\)

(where r = x = 5 m)

= \(\frac{2}{3} \times \frac{22}{7} \times 5^{3} m^{3}\)

= \(\frac{5500}{21} m^{3} \approxeq 261.905 m^{3}\)

= \(262 m^{3} \) (nearest whole number)

(ii) Total surface area of reservoir

= surface area of hemisphere + surface area of cone

= \(2\pi r^{2} + \pi rl\)

From the diagram above, \(l^{2} = 30^{2} + 5^{2}\)

\(l^{2} = 900 + 25 = 925\)

\(l = \sqrt{925} = 30.41 m\)

Hence, the total surface area of the reservoir = \(2 \times \frac{22}{7} \times 5 \times 5 + \frac{22}{7} \times 5 \times 30.41\)

= \(157.143 + 477.871\)

= \(635.104 m^{2}\)

\(\approxeq 635 m^{2}\) (to the nearest whole number).

Explanation

\(\frac{625(\frac{3x}{4} - 1) + 125^{(x - 1)}}{5^{(3x - 2)}}\)

\(\frac{5^4(\frac{3x}{4} - 1) + 5^{3(x - 1)}}{5^{(3x - 2)}}\)

\(\frac{5^{3x - 4} + 5^{3x - 3}}{5^{3x} \times 5^{-2}}\)

= \(\frac{5^{3x}(5^{-4} + 5^{-3})}{5^{3x} \times 5{-2}}\)

= \(\frac{5^{-3}(5^{-1} + 1)}{5^{-2}}\)

= 5\(^{-1} (5^{-1} + 1)\)

= \(\frac{1}{5}(\frac{1}{5} + \frac{1}{1}\))

= \(\frac{1}{5} \times \frac{6}{5} = \frac{6}{25}\)

Correct Answer: B. √13cm

Correct Answer: C. -\(\frac{4}{5}\)

Explanation

tanθ = \(\frac{3}{4}\) → tanθ = 0.75

θ = tan\(^{-1}\)[0.75] → 36.8698°

cosθ = cos[36.8698°]

→ 0.800 or \(\frac{4}{5}\)

However; in the third quadrant Cosine is negative

i.e -\(\frac{4}{5}\)

Correct Answer: D. \(\frac{32}{729}\)

Explanation

ar = \(\frac{2}{9}\) .....(i)

ar\(^3\) = \(\frac{8}{81}\) ......(ii)

\(\frac{ar3}{ar} = \frac{8}{81} \times \frac{9}{2}\)

r\(^2 = \frac{4}{9}\)

r = \(\sqrt{\frac{4}{9}}\)

= \(\frac{2}{3}\)

ar = \(\frac{2}{9}\)

a(\(\frac{2}{3}\)) = \(\frac{2}{9}\)

a = (\(\frac{2}{3}\)) = \(\frac{2}{9}\)

a = \(\frac{2}{9} \times \frac{3}{2}\)

a = \(\frac{1}{3}\)

T\(_r\) = ar\(^5\) = (\(\frac{1}{3}\))(\(\frac{2}{5}\))\(^5\)

= \(\frac{32}{729}\)

Correct Answer: B. \(\frac{{2x+1}^4}{8}\) + C

Explanation

Recall chain rule:

u = 2x +1; du = 2dx → dx = \(\frac{du}{2}\)

u\(^3\) =∫u\(^3\) \(\frac{du}{2}\) → \(\frac{1}{2}\)∫u\(^3\)

= \(\frac{1*u^4}{2*4}\)

= \(\frac{u^4}{8}\) → \(\frac{{2x+1}^4}{8}\) + C

Correct Answer: A. 9

Explanation

\(\frac{x^3 + 6x - 7}{x-1}\):

When numerator is differentiated → 3x\(^2\) + 6

When denominator is differentiated → 1

: \(\frac{3x^2 + 6}{1}\)

substitute x for 1

\(\frac{3 * 1^2 + 6}{1}\) = \(\frac{3 + 6}{1}\)

= \(\frac{9}{1}\)

= 9

Correct Answer: D. (x - 180)°

Correct Answer: A. 3m + n

Explanation

log\(_{10}\) 24 = log\(_{10}\) 8 \(\times\) log\(_{10}\) 3

where log\(_{10}\) 8 = 3 log\(_{10}\) 2 = 3 \(\times\) m

and log\(_{10}\) 3 = n

: log\(_{10}\) 24 = 3m + n

Correct Answer: b. 7hrs