Enrichment, Extension, and Applications

Question 1

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Exercises

  1. Diagram B: Rectangular sides of basket with wire running between opposite midpoints Diagram A: Square prism whose 4 rectangular sides also have wire running between opposite midpoints What will it hold?

    The frame for a rectangular basket is formed from pieces of wire welded together as shown in diagram \( A \), with the following conditions:

    • The basket has an open square-shaped top and a square bottom.
    • Every other face is rectangular, as shown in diagram \( B \).
    • The sides are at right angles to each other and to the base.
    • The total length of wire used is \( 420 \) cm.

    Determine the dimensions which will maximize the volume of the basket.

  2. store (rectangular prism) with x=width of front, a=depth of front, and b=depth of storeroom A wall of sorts...

    A rectangular 'big box' store is to be constructed with a curtain wall running parallel to the front (labelled side \( A \)) and back (labelled side \( B \)) of the store, as shown in the diagram. The curtain wall will divide the store into two areas - a \( 1500 \) square metre sales area in the front, and a \( 500 \) square metre storeroom in the back. Costs are as follows:

    • \( $300 \) per linear metre for the outside walls on the sides and back of the store (note: a linear metre measures length, not area).
    • \( $60 \) per linear metre for the curtain wall.
    • \( $900 \) per linear metre for the front wall, which is mainly glass.

    Find a total cost function, \( C(x) \), for the construction, and hence determine the lengths for \( x \), \( a \), and \( b \) (labelled in diagram) that give the required floor area at minimum cost.

  3. Increasing function, strictly positive, concave up initially, then concave down A worthy investment?

    The graph of the return, \( R(x) \), in dollars, which Monika receives on one of her investments, where \( x \) is the amount she invests is shown.

    Monika claims that she will maximize her return per dollar invested if she invests \( A \) dollars, where \( A \gt 0 \) is the \( x \)-coordinate of the point at which a line passing through the origin \( O \) touches the curve \( y = R(x) \) tangentially.

    Is Monika's reasoning correct? Justify your answer.

  4. How close can you get?
    1. Show that the line \( L: y = 2x - 1 \) does not intersect the curve \( C: y = x^4 + 3x^2 + 2x \).
    2. Let \( P(u, v) \) be the point on the curve \( C \) nearest to the line \( L \). Find the coordinates, \( (u, v) \), of \( P \).

      Hint: Recall that the distance from a point \( P(u, v) \) to the line \( ax + by + c = 0 \) is \( d = \dfrac{\left\lvert au + bv + c \right\rvert}{\sqrt{a^2 + b^2}} \).

    3. How is the slope of the tangent to \( C \) at \( P \) related to the slope of the line \( L \)?
  5. It's in the blood...

    The concentration \( y \) of a certain drug in the bloodstream, \( t \) hours after injection, is given by the function \[ y = \dfrac{at}{1 + \left( \frac{t}{b} \right)^2} \] where \( a \gt 0 \) and \( b \gt 0 \) are constants.

    1. Given that the concentration reaches a maximum of \( 15 \) units after \( 6 \) hours, find the values of \( a \) and \( b \).
    2. Determine the time interval during which the concentration exceeds \( 12 \) units.
  6. Cubic or quadratic - which is greater?

    For two values \( a \) and \( b \), the entity \( M[a, b] \) is defined to the greater of \( a \) or \( b \). Suppose that \begin{align*} g(x) &= -x^3 + 11x^2 - 24x \\ h(x) &= -x^2 + 6x + 8 \\ f(x) &= M\left[ g(x), h(x) \right] \end{align*} on the interval \( 0 \leq x \leq 7, x \in \mathbb{R} \). Find the maximum and minimum values of \( f(x) \) for \( 0 \leq x \leq 7 \), \( x \in \mathbb{R} \).

  7. Extreme circles!
    1. For some \( x \geq 0 \), circle \( C_1 \) has centre \( (x, 0) \) and its circumference passes through \( (1, 1) \).

      For the same \( x \geq 0 \), circle \( C_2 \) has centre \( (x, 0) \) and passes through \( (0, 1) \).

      The function \( R(x) \) is the ratio of the area \( A ( C_1 ) \) to the area \( A ( C_2 ) \). Determine the values of \( x \) for which \( R(x) \) obtains its greatest and least values.

    2. Diagram equilateral triangle and circles C1, C2, C3, and C4 as described in the question

      An equilateral triangle with side length \( 1 \) unit contains three identical circles \( C_1 \), \( C_2 \), and \( C_3 \) of radius \( r_1 \), each touching two sides of the triangle. A fourth circle, \( C_4 \), of radius \( r_2 \), touches each of \( C_1 \), \( C_2 \), and \( C_3 \), as shown. Except for the contact points with \( C_4 \), none of the circles have any points in common with any of the other circles.

      Determine the values of \( r_1 \) and \( r_2 \) which minimize and maximize the sum \( S \) of the areas of the four circles.

      Hint: First, determine how \( r_2 \) is related to \( r_1 \). Then, write \( S \) in terms of \( r_1 \), carefully defining the minimum possible \( r_1 \) and the maximum possible \( r_1 \) by sketching appropriate diagrams for the two cases.

    3. In a certain toy designed for cats, a plastic cone contains a jingling bell which rolls freely inside a clear plastic sphere of radius \( a \) (contained within the cone). Find the maximimum possible volume of the cone in terms of \( a \).
  8. Diagram of y=f(x) as described in question Does \( f(x) \) matter?

    In the given diagram, the graph of \( y = f(x) \) is concave down on \( a \leq x \leq b \). The point \( A\) is at \( (a,0) \) and \( B \) is the point \( (b,0) \). The line \( y=f(x) \) intersects \( x=a \) at \( C \) and intersects \( x=b \) at \( D \). The tangent line at the point \( P \big(p, f(p) \big) \) intersects \( x = a \) at \( Q \) and \( x = b \) at \( R \).

    1. Find the equation of the tangent line at \( P \).
    2. To minimize the area between \( y = f(x) \) and the tangent line \( QR \), it is sufficient to minimize the area \( T(p) \) of the trapezoid \( QABR \).

      1. Find \( T(p) \).
      2. Show that \( T(p) \) has its minimum value for \( p = \dfrac{a + b}{2} \). This shows that the area \( QCDR \) is always minimized when the tangent is located at the midpoint of \( [a, b] \), regardless of what function \( f(x) \) is involved, as long as its graph is concave down on \( [a, b] \).
  9. Diagram of cone and cylinder as described in question A cup of coffee?

    The given diagram represents a conical coffee filter \( 16 \) cm in diameter and \( 24 \) cm high. It drips coffee into a cylindrical pot \( 12 \) cm in diameter.

    When the remaining coffee in the filter is \( 12 \) cm deep, its depth \( y \) is dropping at \( 1 \) cm per minute.

    At that exact time, how fast is the depth \( h \) of coffee in the pot rising?

  10. Diagram as described in question Fountains of colour

    A fountain sprays water symmetrically from the centre of a large pond. Periodically, a fixed quantity of the water is coloured as it emerges, temporarily creating an annulus (a region formed between two concentric circles) of coloured surface water, with fixed area \( A \).

    As the annulus spreads outward, the area \( A_2 \) inside the outer circle \( C_2 \) (radius \( r_2 \)) increases at a rate of \( \pi \) m2 per second.

    What is the rate of change of the circumference of the inner circle \( C_1 \) (radius \( r_1 \)) at the moment when the area \( A_1 \) inside this circle is \( 16\pi \) m2?

  11. Casting shadows
    1. Diagram of situation described in question A street lamp \( 5 \) m tall is \( 6 \) m away from a vertical wall. A \( 2 \) m tall person walks away from the wall, directly towards the street lamp, at a constant speed of \( 1.2 \) m/s.
      1. At what speed is the top of the person's shadow descending down the wall when the person is \( 1.5 \) m from the wall?
      2. After how many seconds will the top of the person's shadow hit the ground?
    2. Diagram of situation described in question

      A streetlight shines atop a \( 10 \) m high vertical pole. A thin flag pole, running parallel to the streetlight, stands \( 10 \) m away. A monkey \( 1 \) m tall climbs the flag pole at a constant rate of \( 2 \) m/s.

      Assuming that the surrounding terrain is flat, how fast is the length of the monkey's shadow growing when it is \( 5 \) m up the flag pole?

    3. Diagram of situation described in question

      A man who is \( 2 \) metres tall walks at \( 1 \) m/s along the edge of a canal that is \( 8 \) metres wide. A floodlight shines from a height of \( 6 \) metres up on the front corner of a building on the opposite side of the canal.

      As the man walks towards the building, what is the rate at which the length of his shadow is changing when he is \( 6 \) metres from the point \( P \), directly across from the light on the other side of the canal?

    4. B: triangle on Cartesian plane with vertices (0,0) (5,10) (-5,10), light at (5,10), bubble up y-axis Diagram A: Martini glass as described with light shining down from the rim into the cup on an angle

      A glass with a cylindrical handle and conical base and cup section is filled with a fizzy beverage. A tiny toy LED light shines from a point along the rim of the glass, as shown in diagram A. As one bubble rises straight up from the bottom of the glass to the top, the light casts the shadow of the bubble onto the opposite wall of the cup section.

      The body of the glass can be modelled as a triangle with height \( 10 \) cm and base \( 10 \) cm, as shown in the diagram B. If the bubble is rising at \( 2 \) cm/s when it is \( 4 \) cm from the bottom of the cup section, how fast is the shadow cast by the bubble moving along the side of the glass at that moment?

  12. Equilateral triangle ABC with particle P part way along side AC and particle Q partway along side BC

    An equilateral triangle \( ABC \) has sides of length \( 60 \) units. One particle \( P \) starts at vertex \( A \) at time \( t = 0 \), and moves along side \( AC \) at a constant rate of \( 2 \) units per second. A second particle \( Q \) starts at the midpoint of \( BC \) and moves along this side at a variable rate such that the line segment \( PQ \) always divides the area of triangle \( ABC \) in half.

    At what time \( t^{*} \) is the particle \( Q \) moving at the same rate as particle \( P \)?

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