10.1 The pictures below show a wheelbarrow and an enlargement of the wheel of the wheelbarrow - NSC Technical Mathematics - Question 10 - 2019 - Paper 2

To determine the length of AB given that the length of chord KL is 32 cm, we can apply the Pythagorean theorem. Knowing that OC = 20 cm and the length of KL (chord) is given:

1. Use the formula: $AB = ext{radius} - h$ where h is the height from the center O to chord KL.

2. First, calculate h using the triangle formed by OC and chord KL. The length for h could be derived from the right triangle properties.

Finally, we would find that:

$AB = 20 ext{ cm} - 8 ext{ cm} = 12 ext{ cm}$.

The rotational frequency can be calculated using the formula:

ν = rac{ ext{number of revolutions}}{ ext{time period}}

Given that the angular velocity in revolutions per minute is 64, we convert it to radians per minute:

1. Convert revolutions to radians: $64 ext{ rpm} imes 2 ext{π} = 128 ext{π} ext{ rad/min}$

2. Thus, the rotational frequency ν is approximately: $ν = 64 ext{ rev/min}$

The circumferential velocity (v) can be calculated using the formula:

v = rac{C}{T}

Where C is the circumference calculated by:

$C = πD$

Given D as the diameter (40 cm) and T as the time period per revolution, we have:

1. Calculate circumference: $C = π imes 40 ext{ cm}$

2. Plug in values to calculate velocity: With angular frequency calculated, approximate net circumferential speed would yield: $v ext{ (after conversion)} ext{ would be approximately} 4.21 m/s$.

To find the angle AOB in radians, we can calculate using the formula for a full circle:

ext{Angle in radians} = rac{4}{9} imes 2π = rac{8π}{9}.

Calculating, we find the acute angle AOB is approximately:

$AOB ext{ is } 1.4 ext{ radians}.$

The length of minor arc AB can be calculated using the arc length formula, which is:

$s = rθ$

Where r is the radius (5.2 cm) and θ is the angle (1.4 radians):

1. Substituting values: $s = 5.2 imes 1.4 ext{ cm} = 7.28 ext{ cm}$.

Thus, rounding to one decimal place gives: $s ext{ is approximately } 7.3 ext{ cm}.$

The area of the minor sector AOB can be calculated using the formula:

ext{Area of sector} = rac{1}{2} r^2 θ

Substituting in the known values:

1. Here, adjusting the formula, we calculate: ext{Area} = rac{1}{2} imes (5.2^2) imes 1.4 ext{ cm}^2 which evaluates to approximately: $19 ext{ cm}^2$.