Q03 Test 31 English

Test No. 31 – Complete Solutions

123

Convert to degrees, minutes, and seconds: (i) 123.456°    (ii) 58.7891°

🔑 Key Concept:
\[ \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600} \] Multiply the decimal part of degrees by 60 → minutes.
Multiply the decimal part of minutes by 60 → seconds.

(i) Convert 123.456° to DMS

Step 1 – Separate whole degrees: Whole degrees \(= 123°\),  Decimal \(= 0.456°\)

Step 2 – Convert decimal degrees to minutes: \[ 0.456 \times 60 = 27.36' \] Whole minutes \(= 27'\),  Decimal \(= 0.36'\)

Step 3 – Convert decimal minutes to seconds: \[ 0.36 \times 60 = 21.6'' \approx 22'' \]

\( \boxed{123.456° = 123°\ 27'\ 22''} \)

(ii) Convert 58.7891° to DMS

Step 1 – Separate whole degrees: Whole degrees \(= 58°\),  Decimal \(= 0.7891°\)

Step 2 – Convert decimal degrees to minutes: \[ 0.7891 \times 60 = 47.346' \] Whole minutes \(= 47'\),  Decimal \(= 0.346'\)

Step 3 – Convert decimal minutes to seconds: \[ 0.346 \times 60 = 20.76'' \approx 21'' \]

\( \boxed{58.7891° = 58°\ 47'\ 21''} \)

124

If arc length is 11 cm and radius \(r = 5\) cm, find the central angle in radians and degrees.

🔑 Key Concept – Central Angle from Arc Length:
The arc length formula is: \[ \ell = r\theta \] Rearranging to find the central angle: \[ \theta = \frac{\ell}{r} \quad \text{(in radians)} \] Then convert to degrees: \(\theta_{\text{deg}} = \theta_{\text{rad}} \times \dfrac{180°}{\pi}\)

Given: Arc length \(\ell = 11\) cm,  Radius \(r = 5\) cm

Step 1 – Find \(\theta\) in radians: \[ \theta = \frac{\ell}{r} = \frac{11}{5} = 2.2\ \text{rad} \]

Step 2 – Convert \(\theta\) to degrees: \[ \theta = 2.2 \times \frac{180°}{\pi} = \frac{2.2 \times 180°}{3.14159} = \frac{396°}{3.14159} \approx 126.05° \]

Step 3 – Convert decimal degrees to DMS: \[ 0.05° \times 60 = 3' \]

\( \boxed{\theta = 2.2\ \text{rad} \approx 126°\ 3'} \)

125

Find the distance between \(C(-5,\,-2)\) and \(D(3,\,2)\).

🔑 Key Concept – Distance Formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Given: \(C(x_1, y_1) = (-5, -2)\),  \(D(x_2, y_2) = (3, 2)\)

Step 1 – Apply the distance formula: \[ d = \sqrt{(3-(-5))^2 + (2-(-2))^2} \]

Step 2 – Simplify inside the root: \[ d = \sqrt{(3+5)^2 + (2+2)^2} = \sqrt{(8)^2 + (4)^2} \] \[ d = \sqrt{64 + 16} = \sqrt{80} \]

Step 3 – Simplify and evaluate: \[ d = \sqrt{16 \times 5} = 4\sqrt{5} \approx 8.94\ \text{units} \]

\( \boxed{d = 4\sqrt{5} \approx 8.94\ \text{units}} \)

126

Find the unknown quantities in the following figures (similar circles).

🔑 Key Concept – Ratio of Areas of Similar Circles:
For two circles with radii \(r_1\), \(r_2\) and areas \(A_1\), \(A_2\): \[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 \] Rearranging to find an unknown radius: \[ r_1 = r_2 \times \sqrt{\frac{A_1}{A_2}} \]

Given: Circle 1: \(r_1 = ?\),  \(A_1 = 153\ \text{cm}^2\)
Circle 2: \(r_2 = 7\ \text{cm}\),  \(A_2 = 833\ \text{cm}^2\)

Step 1 – Write the formula: \[ \frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 \]

Step 2 – Substitute known values: \[ \frac{153}{833} = \left(\frac{r_1}{7}\right)^2 \]

Step 3 – Take the square root of both sides: \[ \frac{r_1}{7} = \sqrt{\frac{153}{833}} = \sqrt{0.18367} \approx 0.4286 \]

Step 4 – Solve for \(r_1\): \[ r_1 = 7 \times 0.4286 \approx 3\ \text{cm} \]

Note – Exact check: \[ \frac{153}{833} = \frac{9}{49} \implies \sqrt{\frac{9}{49}} = \frac{3}{7} \implies r_1 = 7 \times \frac{3}{7} = 3\ \text{cm} \]

\( \boxed{r_1 = 3\ \text{cm}} \)