IB Mathematics Applications & Interpretation AHL 2.9—HL Modelling Functions Questionbank

(g) Compare the evidence standing behind the predicted 16-aisle index with the evidence standing behind the aisle count the model requires for index 50.

A festival has 900 reusable cups at depot A and 600 at depot B. Under its exact planning models, after $t$ days the stocks are $A(t)=900(0.88)^t$ and $B(t)=600(0.94)^t$. Fractional model stocks are permitted, at the complete-day counts as well as between them. The manager wants the first complete day when A has fewer cups than B, and must also arrange for a delivery of A cups to arrive before the first complete-day count at which the original A-stock model would be below 240. An order placed at the start of complete day $n$ arrives at the start of day $n+2$, before that day's stock count.

(a) Show that the ratio $R(t)=A(t)/B(t)$ can be written as $R(t)=1.5\left(\dfrac{44}{47}\right)^{t}$.

(b) Determine the continuous crossover time when the depot stocks are equal.

(c) Compare both stocks on neighboring complete days and determine the first day A is below B.

(d) Calculate the half-life of the ratio and interpret what halves.

A museum keeps a printed reference colour card on the wall of an exhibition room, where the lighting slowly fades it. A conservator measures the card's colour strength so that the card can be replaced before it has faded too far to be used for comparison. The strength after $t$ months on the wall is modelled by $D(t)=ka^{t}$ colour units, where $k\gt0$ and $0\lt a\lt1$. Only two measurements have been made: $D(1)=201.6$ and $D(6)=84.31152758784$. Treat both measurements as exact.

A textile conservator records the colour reading $F(t)$ of a dyed wall hanging, in units, $t$ months after it goes on open display. The first reading is 120 units. The model assumes that the reading approaches a constant background level as time increases. Readings were taken once a month for the first five months only, so the record covers $0\leq t\leq5$. The conservator fits that record with the model $F(t)=70+50(0.82)^t$ and will use the model to decide when the hanging is moved into low-light storage.

(a) Determine the horizontal asymptote and explain why the colour reading itself never falls to half its initial value.

A water company models reservoir depth $H(t)$, in metres, with a repeating yearly cycle, where $t$ is months from the start of a record. The form is $H(t)=a\sin(b(t-c))+d$, with $a,b\gt0$ and $c$ chosen as the smallest positive phase shift consistent with the record. The maximum is 18 m, the minimum is 6 m, the cycle is 12 months, and the first maximum occurs at month 5. All trigonometric arguments are in radians. Restriction measures are active strictly when $H(t)\lt9$ m.

(d) Find both crossings of the 9 m trigger over the cycle $2\leq t\leq14$, using a graph or numerical technology.

(d) Determine the aisle count that the continuous model requires for index 50, and apply the 16-aisle space cap. The manager is also deciding whether adding a fifth aisle to a four-aisle arrangement gives the same gain as a doubling.

(e) Calculate $P(4)$, $P(5)$ and the index gain from 4 to 5.

(f) Explain why the unrestricted mathematical formula has domain $x\gt0$ and no finite upper bound as $x$ increases. Distinguish this from the feasible aisle counts in the warehouse.

(a) Determine the value of $a$.

(b) Determine the value of $k$ and the equation of the fitted model.

(c) Find the half-life of the card's colour strength, in months, and explain why another card with the same monthly decay factor and initial strength $180$ colour units would have the same half-life. The conservator inspects the card only at complete months. The museum's conservation rule is that the card must be replaced as soon as an inspection finds the modelled strength below $65$ colour units.

(d) Determine the time at which the model reaches $65$ colour units, treating $t$ as continuous.

(b) Construct a model $E(t)$ for the excess reading over the background and determine $E(0)$.

(c) Determine the half-life of $E$, in months.

(d) Calculate $F(t)$ after one and two excess half-lives; distinguish these readings from halves of $F(0)$. The conservator inspects the hanging at complete months only. A display rule requires the hanging to be moved at the first complete month for which $F(t)\lt82$.

(a) Determine the amplitude $a$ and principal-axis height $d$.

(b) Determine $b$ and the smallest positive $c$, explaining the first maximum condition.

(c) Construct $H(t)$ and verify its depth at the first maximum and the following minimum.