Exercise 4
The purpose of this exercise is to study the stopping of a cart on a ride, from the moment it enters the braking zone at the end of the course. We denote $t$ the elapsed time, expressed in seconds, from the moment the cart enters the braking zone. We model the distance travelled by the cart in the braking zone, expressed in metres, as a function of $t$, using a function denoted $d$ defined on $[0; +\infty[$. We thus have $d(0) = 0$. Furthermore, we admit that this function $d$ is differentiable on its domain of definition. We denote $d'$ its derivative function.
Part A
In the figure (Fig. 2), we have drawn in an orthonormal coordinate system:

• the representative curve $\mathscr{C}_d$ of the function $d$;
• the tangent $T$ to the curve $\mathscr{C}_d$ at point A with abscissa 4.7;
• the asymptote $\Delta$ to $\mathscr{C}_d$ at $+\infty$.

In this part, no justification is expected. With the precision that the graph allows, answer the questions below. According to this model:

1. After how much time will the cart have travelled 15 m in the braking zone?
2. What minimum length must be provided for the braking zone?
3. What is the value of $d'(4{,}7)$? Interpret this result in the context of the exercise.

Part B
We recall that $t$ denotes the elapsed time, in seconds, from the moment the cart enters the braking zone. We model the instantaneous velocity of the cart, in metres per second ($\mathrm{m.s^{-1}}$), as a function of $t$, by a function $v$ defined on $[0; +\infty[$. We admit that:

• the function $v$ is differentiable on its domain of definition, and we denote $v'$ its derivative function;
• the function $v$ is a solution of the differential equation $$(E): \quad y' + 0{,}6\, y = \mathrm{e}^{-0{,}6t},$$ where $y$ is an unknown function and $y'$ is the derivative function of $y$.

We further specify that, upon arrival in the braking zone, the velocity of the cart is equal to $12\,\mathrm{m.s^{-1}}$, that is $v(0) = 12$.

1. 1. [a.] We consider the differential equation $$(E'): \quad y' + 0{,}6\, y = 0$$ Determine the solutions of the differential equation $(E')$ on $[0; +\infty[$.
   2. [b.] Let $g$ be the function defined on $[0; +\infty[$ by $g(t) = t\,\mathrm{e}^{-0{,}6t}$. Verify that the function $g$ is a solution of the differential equation $(E)$.
   3. [c.] Deduce the solutions of the differential equation $(E)$ on $[0; +\infty[$.
   4. [d.] Deduce that for every real $t$ belonging to the interval $[0; +\infty[$, we have: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}$$
2. In this question, we study the function $v$ on $[0; +\infty[$.
   1. [a.] Show that for every real $t \in [0; +\infty[$, $v'(t) = (-6{,}2 - 0{,}6t)\,\mathrm{e}^{-0{,}6t}$.
   2. [b.] By admitting that: $$v(t) = 12\,\mathrm{e}^{-0{,}6t} + \frac{1}{0{,}6} \times \frac{0{,}6t}{\mathrm{e}^{0{,}6t}}$$ determine the limit of $v$ at $+\infty$.
   3. [c.] Study the direction of variation of the function $v$ and draw up its complete variation table. Justify.
   4. [d.] Show that the equation $v(t) = 1$ has a unique solution $\alpha$, of which you will give an approximate value to the nearest tenth.
3. When the velocity of the cart is less than or equal to 1 metre per second, a mechanical system is triggered allowing its complete stopping. Determine after how much time this system comes into action. Justify.

Part C
We recall that for every real $t$ belonging to the interval $[0; +\infty[$: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}.$$ We admit that for every real $t$ in the interval $[0; +\infty[$: $$d(t) = \int_0^t v(x)\,\mathrm{d}x$$

1. Using integration by parts, show that the distance travelled by the cart between times 0 and $t$ is given by: $$d(t) = \mathrm{e}^{-0{,}6t}\left(-\frac{5}{3}t - \frac{205}{9}\right) + \frac{205}{9}$$
2. We recall that the stopping device is triggered when the velocity of the cart is less than or equal to 1 metre per second. Determine, according to this model, an approximate value to the nearest hundredth of the distance travelled by the cart in the braking zone before the triggering of this device.

Let P be a point on the curve $y = 2 e ^ { - x }$ at $\mathrm { P } \left( t , 2 e ^ { - t } \right)$ $(t > 0)$. Let A be the foot of the perpendicular from P to the $y$-axis, and let B be the point where the tangent line at P intersects the $y$-axis. What is the value of $t$ that maximizes the area of triangle APB? [4 points]
(1) 1
(2) $\frac { e } { 2 }$
(3) $\sqrt { 2 }$
(4) 2
(5) $e$

The derivative $f'(x)$ of a function $f(x)$ that is differentiable on the set of all real numbers is $$f'(x) = -x + e^{1 - x^{2}}$$ For a positive number $t$, let $g(t)$ be the area of the region enclosed by the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$, the curve $y = f(x)$, and the $y$-axis. What is the value of $g(1) + g'(1)$? [4 points]
(1) $\frac{1}{2}e + \frac{1}{2}$
(2) $\frac{1}{2}e + \frac{2}{3}$
(3) $\frac{1}{2}e + \frac{5}{6}$
(4) $\frac{2}{3}e + \frac{1}{2}$
(5) $\frac{2}{3}e + \frac{2}{3}$

18. (Total Score: 12 points)
A unit uses wood to make a frame as shown in the figure. The lower part of the frame is a rectangle with sides $x$ and $y$ (in meters). The upper part is an isosceles right triangle. The total area enclosed by the frame is required to be $8 \text{ m}^2$. Find the values of $x$ and $y$ (accurate to 0.001 m) that minimize the amount of material used. [Figure]

20. (Total Score: 14 points) Subproblem 1: 7 points, Subproblem 2: 7 points.
As shown in the figure, to make a cylindrical lantern, 4 congruent rectangular frames are first made, using a total of 9.6 meters of wire. Then $S$ square meters of plastic sheet is used to form the lateral surface and bottom of the cylinder (the top is not installed). [Figure]
(1) For what value of the cylinder's base radius $r$ does $S$ attain its maximum value? Find this maximum value (result accurate to 0.01 square meters);
(2) To make a lantern as shown with base radius 0.3 meters, draw the three-view drawing for making the lantern (when drawing, structural factors such as frames need not be considered).

10. If a function $f ( x )$ defined on $\mathbb{R}$ satisfies $f ( 0 ) = - 1$, and its derivative $f ^ { \prime } ( x )$ satisfies $f ^ { \prime } ( x ) > k > 1$, then among the following conclusions, the one that must be wrong is
A. $f \left( \frac { 1 } { k } \right) < \frac { 1 } { k }$
B. $f \left( \frac { 1 } { k } \right) > \frac { 1 } { k - 1 }$
C. $f \left( \frac { 1 } { k - 1 } \right) < \frac { 1 } { k - 1 }$
D. $f \left( \frac { 1 } { k - 1 } \right) > \frac { k } { k - 1 }$

Section II (Non-Multiple Choice Questions, 100 points)

II. Fill-in-the-Blank Questions: This section contains 5 questions, each worth 4 points, for a total of 20 points. Write your answers in the corresponding positions on the answer sheet.

A sphere $O$ has radius 1. A pyramid has its apex at $O$ and the four vertices of its base all on the surface of sphere $O$. When the volume of this pyramid is maximized, its height is
A. $\frac { 1 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

For $11 \leq t \leq 25$, the cooling process of the mixture can be approximated by the function $u _ { 2 }$ defined on $\mathbb { R }$ with $u _ { 2 } ( t ) = 18 + 36 \cdot \mathrm { e } ^ { - 0,033 \cdot t }$.
Here, $t$ again denotes the time in minutes since the beginning of the investigation and $u _ { 2 } ( t )$ denotes the temperature of the mixture in ${ } ^ { \circ } \mathrm { C }$. The graph of $u _ { 2 }$ is monotonically decreasing.
Determine the average rate of change of the temperature of the coffee in the first 10 minutes of the investigation and the average rate of change of the temperature of the mixture from the 11th to the 21st minute of the investigation.
Compare the results.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Deduce that $\psi$ is differentiable on $\mathbb{R}_+^*$ and that $m(h) = \frac{u(h) - h}{\beta}$ for all $h \in \mathbb{R}_+^*$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Show then that $m^+ = 0$ if $\beta \leqslant 1$, and $m^+ > 0$ if $\beta > 1$.

121. In triangle ABC, $BC = 20$, $AH = 12$, and $AB = 12$ units. Line $\Delta$ parallel to BC moves at a constant speed of $0.2$ units per second. The rate of increase of the area of the trapezoid at the moment when the distance between the two parallel lines is 9 units is which of the following?
[Figure: Triangle ABC with vertex A at top, base BC at bottom, height AH drawn, points D and E on sides AB and AC respectively forming a trapezoid BCED, with H the foot of the altitude on BC]
(1) $0.8$ [4pt] (2) $0.9$ [4pt] (3) $1$ [4pt] (4) $1.2$
%% Page 23

121- In manufacturing a cone in the shape of a right circular cone with volume $\dfrac{\pi}{3}$, at what height is the least material consumed?
(1) $\dfrac{\sqrt{2}}{2}$ (2) $1$ (3) $\sqrt[4]{2}$ (4) $\sqrt{2}$

Let $a$, $b$, $h$ be the three edges meeting at a particular vertex of a triangular prism, such that $a$, $b$ are sides of a base triangle with angle $\theta$ between them and $h$ is the height of the prism. Given that the total surface area is $K$, show that the volume $V$ satisfies $V \leq \sqrt{K^3/54}$, and find the dimensions of the prism of maximum volume.

A triangle has vertices $A$, $B$, $C$. A point $P$ is chosen on side $AB$, and lines through $P$ parallel to the other sides create smaller triangles $APQ$ and $BPR$ and a parallelogram $PQCR$. Find the minimum value of the maximum of the areas of triangles $APQ$ and $BPR$ as a fraction of the area of $ABC$.

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

A lantern is placed on the ground 100 feet away from a wall. A man six feet tall is walking at a speed of 10 feet/second from the lantern to the nearest point on the wall. When he is midway between the lantern and the wall, the rate of change (in ft./sec.) in the length of his shadow is
(A) 2.4
(B) 3
(C) 3.6
(D) 12

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

A circular lawn of diameter 20 meters on a horizontal plane is to be illuminated by a light-source placed vertically above the centre of the lawn. It is known that the illuminance at a point $P$ on the lawn is given by the formula $I = \frac{C \sin\theta}{d^2}$ for some constant $C$, where $d$ is the distance of $P$ from the light-source and $\theta$ is the angle between the line joining the centre of the lawn to $P$ and the line joining the light-source to $P$. Then the maximum possible illuminance at a point on the circumference of the lawn is
(A) $\frac{C}{75\sqrt{3}}$
(B) $\frac{C}{100\sqrt{3}}$
(C) $\frac{C}{150\sqrt{3}}$
(D) $\frac{C}{250\sqrt{3}}$.

A line $L : y = m x + 3$ meets $y$-axis at $E ( 0,3 )$ and the arc of the parabola $y ^ { 2 } = 16 x$, $0 \leq y \leq 6$ at the point $F \left( x _ { 0 } , y _ { 0 } \right)$. The tangent to the parabola at $F \left( x _ { 0 } , y _ { 0 } \right)$ intersects the $y$-axis at $G \left( 0 , y _ { 1 } \right)$. The slope $m$ of the line $L$ is chosen such that the area of the triangle $E F G$ has a local maximum.
Match List I with List II and select the correct answer using the code given below the lists:
List I

• [P.] $m =$
• [Q.] Maximum area of $\triangle EFG$ is
• [R.] $y_0 =$
• [S.] $y_1 =$

List II

1. $\frac{1}{2}$
2. $4$
3. $2$
4. $1$

Codes:

	P	Q	R	S
(A)	4	1	2	3
(B)	3	4	1	2
(C)	1	3	2	4
(D)	1	3	4	2

For $x \in \mathbb { R }$, let $\tan ^ { - 1 } ( x ) \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$. Then the minimum value of the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \int _ { 0 } ^ { x \tan ^ { - 1 } x } \frac { e ^ { ( t - \cos t ) } } { 1 + t ^ { 2023 } } d t$ is

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement x is proportional to
(1) $x ^ { 3 }$
(2) $e ^ { x }$
(3) $x$
(4) $\log _ { e } x$

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f ( x ) = \log _ { \mathrm { e } } x$ on the interval $[ 1,3 ]$ is
(1) $2 \log _ { 3 } e$
(2) $\frac { 1 } { 2 } \log _ { e } 3$
(3) $\log _ { 3 } e$
(4) $\log _ { e } 3$

A spherical balloon is filled with 4500$\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is
(1) $\frac{9}{7}$
(2) $\frac{7}{9}$
(3) $\frac{2}{9}$
(4) $\frac{9}{2}$

A body starts from rest on a long inclined plane of slope $45^{\circ}$. The coefficient of friction between the body and the plane varies as $\mu = 0.3x$, where $x$ is distance travelled down the plane. The body will have maximum speed (for $g = 10\mathrm{~m/s^2}$) when $x =$
(1) 9.8 m
(2) 27 m
(3) 12 m
(4) 3.33 m

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is
(1) 3500
(2) 4500
(3) 2500
(4) 3000