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The depth of water in a harbor, in meters, is modeled by
D(t)=4+2sin(6πt)
where t is the number of hours after midnight.
- [1]
Write down the mean depth of the water.
- [1]
Find the maximum depth of the water.
- [1]
Find the depth of the water at 2AM.
- [2]
State the period of the function and explain what it represents in this context.
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Hugh notices that the average daily temperature T in his town is modeled by
T(t)=15+8sin(12π(t−6))°C,
where t is the hours after midnight.
- [1]
Find the domain of T.
- [2]
Find the range of T.
- [2]
Determine the average temperature in Hugh's town when he leaves for work at 8:30 AM.
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The points P and Q lie on a circle with center O and radius r such that PO^Q=2 radians.
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The perimeter of the shaded region is 7π.
- [4]
Find the value of r.
- [2]
Hence find the area of the region inside the circle that is not shaded.
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A spherical white blood cell appears as a circle with circumference 6π×10−5m.
- [1]
Write down the radius of the cell.
- [3]
Find the volume of the cell, giving your answer in the form π(a×10k)m3, where a<10 and k∈Z.
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A chocolate ball is formed from a spherical shell with diameter d and thickness 2mm. The shell is filled on the inside with 3cm3 of caramel.
- [3]
Show that d=2.19cm.
- [3]
Hence find the volume of the chocolate shell.
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The diagram below shows triangle ABC
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- [4]
Find the value of x.
- [2]
Hence or otherwise, find the value of the angle θ to the nearest degree.
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The blades of a wind turbine have a diameter of 16m and rotate clockwise at a constant speed, 1 revolution every 4 seconds. The blades are fixed on a shaft such that the tips of the blades are always at least 7m above the ground. The point Q lies at the tip of one of the blades.
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Let h be the height, in meters, of Q above the ground. After t minutes, h is given by h(t)=acos(bt)+c, where a,b,c∈R and a>0.
- [2]
Show that Q starts at the highest possible point.
- [4]
Find the values of a, b and c.
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The following diagram shows a ferris wheel, which rotates at a constant rate.
The height h, in meters, of point P above the ground is given by h(t)=20cos(bt)+23, where t is the time in minutes.
Find
- [1]
the distance from the lowest point on the wheel to the ground,
- [1]
the diameter of the wheel.
The wheel makes 4 revolutions in an hour.
- [3]
Find the value of b.
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