A city with a population of 800,000 has been growing at a rate of 7% per year. If this rate continues, find the population of th
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Answer:
a) see below
b) radius = 16.4 in (1 d.p.)
c) 18°. Yes contents will remain. No, handle will not rest on the ground.
d) Yes contents would spill. Max height of handle = 32.8 in (1 d.p.)
Step-by-step explanation:
<u>Part a</u>
A chord is a <u>line segment</u> with endpoints on the <u>circumference</u> of the circle.
The diameter is a <u>chord</u> that passes through the center of a circle.
Therefore, the spokes passing through the center of the wheel are congruent chords.
The spokes on the wheel represent the radii of the circle. Spokes on a wheel are usually evenly spaced, therefore the congruent central angles are the angles formed when two spokes meet at the center of the wheel.
<u>Part b</u>
The <u>tangent</u> of a circle is always <u>perpendicular</u> to the <u>radius</u>.
The tangent to the wheel touches the wheel at point B on the diagram. The radius is at a right angle to this tangent. Therefore, we can model this as a right triangle and use the <u>tan trigonometric ratio</u> to calculate the radius of the wheel (see attached diagram 1).
where:
is the angle
- O is the side opposite the angle
- A is the side adjacent the angle
Given:
- O = radius (r)
- A = 45 in
Substituting the given values into the tan trig ratio:
Therefore, the radius is 16.4 in (1 d.p.).
<u>Part c</u>
The measure of an angle formed by a secant and a tangent from a point outside the circle is <u>half the difference</u> of the measures of the <u>intercepted arcs</u>.
If the measure of the arc AB was changed to 72°, then the other intercepted arc would be 180° - 72° = 108° (since AC is the diameter).
As the handle of the cart needs to be no more than 20° with the ground for the contents not to spill out, the contents will remain in the handcart at an angle of 18°.
The handle will not rest of the ground (see attached diagram 2).
<u>Part d</u>
This can be modeled as a right triangle (see diagram 3), with:
- height = (48 - r) in
- hypotenuse ≈ 48 in
Use the sin trig ratio to find the angle the handle makes with the horizontal:
As 41.2° > 20° the contents will spill out the back.
To find the <u>maximum height</u> of the handle from the ground before the contents start spilling out, find the <u>height from center of the wheel</u> (setting the angle to its maximum of 20°):
Then add it to the radius:
(see diagram 4)
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<u>Circle Theorem vocabulary</u>
<u>Secant</u>: a straight line that intersects a circle at two points.
<u>Arc</u>: the curve between two points on the circumference of a circle
<u>Intercepted arc</u>: the curve between the two points where two chords or line segments (that meet at one point on the other side of the circle) intercept the circumference of a circle.
<u>Tangent</u>: a straight line that touches a circle at only one point.
