Math  /  Algebra

Question2. Eduardo's cat runs inside a circular exercise wheel, making the wheel spin at a constant rate in an anticlockwise direction. The height, h cmh \mathrm{~cm}, of a fixed point, P , on the wheel can be modelled by h(t)=asin(bt)+ch(t)=\operatorname{asin}(b t)+c where tt is the time in seconds and a,ba, b, cZ+\mathrm{c} \in \mathrm{Z}+. When t=0t=0, point PP is at a height of 78 cm . (a) Write down the value of c .
When t=4t=4, point PP first reaches its maximum height of 143 cm . (b) Find the value of [1] (i) a. (ii) b. (c) Write down the minimum height of point P . [3] [1]
Later, the cat is tired, and it takes twice as long for point P to complete one revolution at a new constant rate. (d) Write down the new value of bb. [1]

Studdy Solution

STEP 1

What is this asking? We're figuring out how high a point on a spinning cat wheel goes at different times, and how the height changes when the wheel spins slower. Watch out! Remember that sine goes up and down, so we need to think about maximum and minimum values carefully!
Also, make sure we're working with radians for *b*, not degrees.

STEP 2

1. Find *c*
2. Find *a*
3. Find *b*
4. Find the minimum height
5. Find the new *b*

STEP 3

We're given the height formula h(t)=asin(bt)+ch(t) = a \cdot \sin(b \cdot t) + c.
We know that when t=0t = 0, the height h(0)h(0) is **78 cm**.
Let's plug that in!

STEP 4

So, h(0)=asin(b0)+c=78h(0) = a \cdot \sin(b \cdot 0) + c = 78.
Since sin(0)=0\sin(0) = 0, this simplifies to a0+c=78a \cdot 0 + c = 78, which means c=78c = 78!

STEP 5

We know that the **maximum height** is **143 cm** when t=4t = 4.
We also know now that c=78c = 78.
Let's use this!

STEP 6

Our formula is now h(t)=asin(bt)+78h(t) = a \cdot \sin(b \cdot t) + 78.
At t=4t = 4, we have h(4)=asin(b4)+78=143h(4) = a \cdot \sin(b \cdot 4) + 78 = 143.

STEP 7

Since 143 is the **maximum height**, the sine part must be at its maximum value, which is **1**!
So, a1+78=143a \cdot 1 + 78 = 143.

STEP 8

Subtracting 78 from both sides gives us a=14378=65a = 143 - 78 = 65.
Awesome!

STEP 9

We know a=65a = 65 and c=78c = 78, and the **maximum height** is reached when t=4t = 4.
This means a quarter of a full sine wave happens in 4 seconds.

STEP 10

A full sine wave takes 44=164 \cdot 4 = 16 seconds.
This is the **period** of our sine function.

STEP 11

The relationship between the period and *b* is given by Period=2πb \text{Period} = \frac{2\pi}{b} .
We know the period is **16**, so 16=2πb16 = \frac{2\pi}{b}.

STEP 12

Multiplying both sides by *b* gives 16b=2π16 \cdot b = 2\pi.
Dividing both sides by 16 gives b=2π16=π8b = \frac{2\pi}{16} = \frac{\pi}{8}.
Perfect!

STEP 13

The minimum value of sin(bt)\sin(b \cdot t) is **-1**.
So, the minimum height is h(t)=65(1)+78h(t) = 65 \cdot (-1) + 78.

STEP 14

This simplifies to h(t)=65+78=13h(t) = -65 + 78 = 13.
So, the minimum height is **13 cm**.

STEP 15

The cat is tired!
Now it takes *twice* as long for one revolution.
So, the new period is 162=3216 \cdot 2 = 32 seconds.

STEP 16

Using the period formula again, 32=2πnew b32 = \frac{2\pi}{\text{new } b}.

STEP 17

Multiplying both sides by the new *b* gives 32new b=2π32 \cdot \text{new } b = 2\pi.
Dividing both sides by 32 gives new b=2π32=π16\text{new } b = \frac{2\pi}{32} = \frac{\pi}{16}.

STEP 18

(a) c=78c = 78 (b) (i) a=65a = 65 (ii) b=π8b = \frac{\pi}{8} (c) Minimum height = 13 cm (d) New b=π16b = \frac{\pi}{16}

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