$1$ . a) John Doe is tied to an elastic cord and is suspended in the air at rest $50\,\text{m}$ above the ground. He grabs the cord and raises himself $4\,\text{m}$ up at time $t=0$ . If he let go of the rope, he returns to the highest position in $1.2\,\text{s}$ , determine the height of John Doe above the ground at $t=0.7\,\text{s}$ rounded to $2$ decimal places. b) Sketch a nicely labelled cosine graph for $2$ cycles.

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Find probabilities using the binomial distribution

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1

. a) John Doe is tied to an elastic cord and is suspended in the air at rest

50\,\text{m}

above the ground. He grabs the cord and raises himself

4\,\text{m}

up at time

t=0

. If he let go of the rope, he returns to the highest position in

1.2\,\text{s}

, determine the height of John Doe above the ground at

t=0.7\,\text{s}

rounded to

2

decimal places. b) Sketch a nicely labelled cosine graph for

2

cycles.

Concept of Simple Harmonic Motion: To find the height at $t=0.7$ s, we can use the concept of simple harmonic motion, which can be modeled by a cosine function. The maximum height is $54$ m, which is the amplitude of the cosine function.
Period of the Motion: The period of the motion is the time it takes to return to the highest position, which is given as $1.2$ seconds for half a cycle. Therefore, the full cycle period is $2 \times 1.2 = 2.4$ seconds.
Cosine Function for Height: The cosine function for the height as a function of time can be written as $h(t) = A \cdot \cos(B \cdot t) + C$ , where $A$ is the amplitude, $B$ is related to the period, and $C$ is the vertical shift.
Calculating B: We know $A = 54\,\text{m}$ , $C = 50\,\text{m}$ , and the period $T = 2.4\,\text{s}$ . To find $B$ , we use the formula $B = \frac{2\pi}{T}$ . So $B = \frac{2\pi}{2.4}$ .
Function Evaluation at $t=0.7\,\text{s}$ : Calculating $B$ gives us $B = \frac{2\pi}{2.4} \approx 2.618$ . Now we have the function $h(t) = 54 \cdot \cos(2.618 \cdot t) + 50$ .
Calculation of $h(0.7)$ : To find the height at $t=0.7$ s, we plug in the value of $t$ into the function: $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50$ .
Calculation of h( $0$ . $7$ ): To find the height at $t=0.7$ s, we plug in the value of $t$ into the function: $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50$ . Calculating $h(0.7)$ gives us $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50 \approx 54 \times \cos(1.8326) + 50$ .
Calculation of $h(0.7)$ : To find the height at $t=0.7$ s, we plug in the value of $t$ into the function: $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50$ .Calculating $h(0.7)$ gives us $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50 \approx 54 \times \cos(1.8326) + 50$ .Using a calculator, $\cos(1.8326) \approx -0.34$ (rounded to two decimal places). So, $h(0.7) \approx 54 \times (-0.34) + 50$ .
Calculation of $h(0.7)$ : To find the height at $t=0.7$ s, we plug in the value of $t$ into the function: $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50$ . Calculating $h(0.7)$ gives us $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50 \approx 54 \times \cos(1.8326) + 50$ . Using a calculator, $\cos(1.8326) \approx -0.34$ (rounded to two decimal places). So, $h(0.7) \approx 54 \times (-0.34) + 50$ . Calculating $h(0.7)$ gives us $h(0.7) \approx -18.36 + 50$ .
Calculation of h( $0$ . $7$ ): To find the height at $t=0.7$ s, we plug in the value of $t$ into the function: $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50$ . Calculating $h(0.7)$ gives us $h(0.7) = 54 \times \cos(2.618 \times 0.7) + 50 \approx 54 \times \cos(1.8326) + 50$ . Using a calculator, $\cos(1.8326) \approx -0.34$ (rounded to two decimal places). So, $h(0.7) \approx 54 \times (-0.34) + 50$ . Calculating $h(0.7)$ gives us $h(0.7) \approx -18.36 + 50$ . Adding up, we get $h(0.7) \approx 31.64$ m.