Solved: The Texas Star is a Ferris wheel at the state fair in Dallas. The Ferris wheel... [IB Mathematics Applications & Interpretation (Math AI)]

Question

SLPaper 2

The Texas Star is a Ferris wheel at the state fair in Dallas. The Ferris wheel has a diameter of $61.8 m$ . To begin the ride, a passenger gets into a chair at the lowest point on the wheel, which is $2.7 m$ above the ground, as shown in the following diagram. A ride consists of multiple revolutions, and the Ferris wheel makes $1.5$ revolutions per minute.

Theheight of a chair above the ground, $h$ , measured in metres, during a ride on the Ferris wheel can be modelled by the function $h (t) = - a \cos (b t) + d$ , where $t$ is the time, in seconds, since a passenger began their ride.

Calculate the value of

A ride on the Ferris wheel lasts for $12$ minutes in total.

For exactly one ride on the Ferris wheel, suggest

Big Tex is a $16.7$ metre-tall cowboy statue that stands on the horizontal ground next to theFerris wheel.

^{[Source: Aline Escobar., n.d. Cowboy. [image online] Available at: https://thenounproject.com/search/?q=cowboy&i=1080130
This fileis licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
https://creativecommons.org/licenses/by-sa/3.0/deed.en [Accessed 13/05/2021]. Source adapted.]}

There is a plan to relocate the Texas Star Ferris wheel onto a taller platform which will increasethe maximum height of the Ferris wheel to $65.2 m$ . This will change the value of one parameter, $a$ , $b$ or $d$ , found in part (a).

$a$ .

[2]

Verified

Solution

an attempt to find the amplitude (M1)

$\frac{61.8}{2}$ OR $\frac{64.5 - 2.7}{2}$

$(a =) 30.9 m$ A1

Note: Accept an answer of $(a =) - 30.9 m$ .

[2 marks]

$b$ .

[2]

Verified

Solution

(period $= \frac{60}{1.5} =$ ) $40 (s)$ (A1)

( $(b =) \frac{360 °}{40}$ )

$(b =) 9$ A1

Note:Accept an answer of $(b =) - 9$ .

[2 marks]

$d$ .

[2]

Verified

Solution

attempt to find $d$ (M1)

$(d =) 30.9 + 2.7$ OR $\frac{64.5 + 2.7}{2}$

$(d =) 33.6 m$ A1

[2 marks]

Calculate the number of revolutions of the Ferris wheel per ride.

[2]

Verified

Solution

$12 \times 1.5$ OR $\frac{12 \times 60}{40}$ (M1)

$18$ (revolutions per ride) A1

[2 marks]

an appropriate domain for $h (t)$ .

[1]

Verified

Solution

$0 \leq t \leq 720$ A1

[1 mark]

an appropriate range for $h (t)$ .

[2]

Verified

Solution

$2.7 \leq h \leq 64.5$ A1A1

Note: Award A1 for correct endpoints of domain and A1 for correct endpoints of range. Award A1 for correct direction of both inequalities.

[2 marks]

By considering the graph of $h (t)$ ,determine the length of time during one revolution ofthe Ferris wheel for which the chair is higher than the cowboy statue.

[3]

Verified

Solution

graph of $h (t)$ and $y = 16.7$ OR $h (t) = 16.7$ (M1)

$6.31596 \dots$ and $33.6840 \dots$ (A1)

$27.4 (s) (27.3680 \dots)$ A1

[3 marks]

Identify which parameter will change.

[1]

Verified

Solution

$d$ A1

[1 mark]

Find the new value of the parameter identified in part (e)(i).

[2]

Verified

Solution

EITHER

$d + 30.9 = 65.2$ (A1)

$65.2 - (61.8 + 2.7) = 0.7$ (A1)

$3.4$ (new platform height) (A1)

THEN

$(d =) 34.3 m$ A1

[2 marks]

Mathematics Applications & Interpretation (Math AI) International Baccalaureate (IB) Practice Question: The Texas Star is a Ferris wheel at the state fair in Dallas. The Ferris wheel has a diameter of 61.8 m. To begin the ri...