Using mathematical tables, find the reciprocal of 0.5641, hence evaluate sqrt(0.0441)0.5641. In the diagram below, XY is a transverse tangent to the circles whose centres are A and B with radii 20cm a

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1. Using mathematical tables, find the reciprocal of 0.5641, hence evaluate $\frac{\sqrt{0.0441}}{0.5641}$.

• Step 1: Find the reciprocal of 0.5641. The reciprocal of 0.5641 is $\frac{1}{0.5641}$. Using a calculator (or mathematical tables for reciprocals):

  $$\frac{1}{0.5641} \approx 1.772735$$

• Step 2: Find the square root of 0.0441. Using a calculator (or mathematical tables for square roots):

  $$\sqrt{0.0441} = 0.21$$

• Step 3: Evaluate the expression. Substitute the values found in Step 1 and Step 2 into the expression:

  $$\frac{\sqrt{0.0441}}{0.5641} = \frac{0.21}{0.5641}$$

  This can also be written as $0.21 \times \frac{1}{0.5641}$.

  $$0.21 \times 1.772735 \approx 0.372274$$

  Rounding to four decimal places:

  $$\frac{0.21}{0.5641} \approx 0.3723$$

  The final answer is $\boxed{0.3723}$

2. In the diagram below, XY is a transverse tangent to the circles whose centres are A and B with radii 20cm and 12cm respectively. The distance between centres A and B is 40cm. Calculate the length of tangent XY.

• Step 1: Identify the given values. Radius of circle A, $R_1 = 20$ cm. Radius of circle B, $R_2 = 12$ cm. Distance between centres A and B, $D = 40$ cm. We need to find the length of the transverse common tangent, $L = XY$.

• Step 2: Use the formula for the length of a transverse common tangent. The formula for the length of a transverse common tangent is:

  $$L = \sqrt{D^2 - (R_1 + R_2)^2}$$

• Step 3: Substitute the values and calculate.

  $$L = \sqrt{40^2 - (20 + 12)^2}$$ $$L = \sqrt{40^2 - 32^2}$$ $$L = \sqrt{1600 - 1024}$$ $$L = \sqrt{576}$$ $$L = 24 \text{ cm}$$

  The final answer is $\boxed{24 \text{ cm}}$

3. The figure below shows a right pyramid on a square base of side 10cm. The slant length of the pyramid is 13cm. X is the mid-point of RS and OY is the perpendicular height of the pyramid. Calculate the height of the pyramid.

• Step 1: Identify the given information from the figure and text. The base PQRS is a square with side length = 10 cm. The slant edge YS = 13 cm (as indicated in the diagram). OY is the perpendicular height of the pyramid. O is the center of the square base.

• Step 2: Calculate the length of OS. O is the center of the square base. OS is half the length