IB Mathematics Applications & Interpretation SL 2.6—modelling Skills Questionbank

A harbour’s water level is modeled over a 12-hour cycle. Time $t=0$ is defined at high water, and angles are measured in degrees. The maximum level is $10$ m, the minimum is $4$ m, and the principal axis is crossed after one quarter-cycle. The harbour master uses a cosine model with no phase shift, as required by this time origin. A vessel may enter only when the modeled level is greater than $8$ m.

(a) Determine the amplitude, principal-axis equation and period.

(c) Determine, using graphing technology in degree mode, both threshold crossings in one cycle.

(d) State the intervals in the cycle during which entry is permitted and calculate their total duration.

(e) Find the total permitted duration in 24 hours and explain the recurring window.

A stadium models demand $D$ for drinks at price $p$ euros. Exact pilot values are $D(4)=440$ and $D(9)=240$, and the manager assumes a linear demand model. Revenue is $R(p)=pD(p)$ euros. Only prices and demands that are nonnegative are considered. The manager must set a trial price and will order stock from the demand implied by that choice. Report money to two decimal places.

(a) Determine the linear demand model and interpret its gradient.

(b) Find the revenue model and its reasonable price domain.

(c) Determine, using graphing technology, the maximum revenue and interpret the vertex.

A logistics company provides three types of parcel delivery services: Standard, Express, and Overnight. The costs per parcel for each service are shown in the table below.

On a specific day:

• A total of 250 parcels were delivered.
• The total revenue collected from these deliveries was $2010.
• There were 30 more Standard deliveries than Overnight deliveries.

Let $x, y,$ and $z$ represent the number of Standard, Express, and Overnight deliveries respectively.

(a) Write down a system of three equations representing the information above.

(b) Calculate the number of each type of delivery made on that day.

In this question, provide all final values rounded to two decimal places.

A decorator is repainting a community hall. One litre of the wall paint covers 6.8 m$^2$. The hall has 145 m$^2$ of wall area to paint. The decorator buys 7% extra paint to allow for wastage.

For a mural, a different paint is used. The area covered is modelled by $A = Ly$, where $A$ is the area in m$^2$ and $L$ is the number of litres of paint used. The decorator uses 18.5 L of paint to cover 127.31 m$^2$.

(b) Calculate the total quantity of paint the decorator buys, inclusive of the wastage allowance.

(b) Determine the model in the form $H(t)=a\cos(bt)+d$, with its argument in degrees.

(f) Comment on a printed tide table that shows whole hours only, so that each crossing time appears rounded to the nearest hour and a skipper reads the entry window from those two printed hours alone. In your comment, address which way each end of the window moves, what that means for a vessel entering the harbour, and whether the nearest-hour rule could be trusted at a different entry level.

(d) Calculate the revenues at the two pilot prices and compare them with the maximum.

A fixture is financially viable under the model when revenue is at least $2000.00$ euros.

(a) Determine the quantity of paint, in litres, needed to cover 145 m$^2$, based on the provided coverage rate.

(c) Determine the value of $y$.