Q1 It costs a baker a fixed cost of $420 and a variable cost of $2.10 per cupcake. A cupcake is sold for $4.90 each. (i) Make a table showing the cost of producing 20,40,60,80 and 100 cupcakes. (ii) Make a table showing the revenue from selling 20,40,60,80 and 100 cupcakes. (iii) Write an algebraic expression representing the cost C as a function of the number of cupcakes x that are produced. (iv) Write an algebraic expression representing the revenue R as a function of the number of cupcakes x sold. (v) Graph both functions on the same coordinate axes. (vi) From your graph find coordinate at which cost equals revenue. (vii) Using your graph, determine how many cupcakes need to be made to produce revenue of at least $1,029.How much profit is made for this number of cupcakes? ? ?. (i) Discuss the continuity of the function f(x) = {x2 − 3x if x < 2 2x + 4 if x ≥ 2 on the open interval 0 < x < 2 and on the closed interval 0 ≤ x ≤ 2. (ii) An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M.will have assembled f (x) = x3 + 6x2 + 15x units x hours later. Approximately how many units will the worker assemble between 8:00 and 8:15 A.M.? (iii) A manufacturer’s total cost is C(q) = 0.1q3 + 0.5q2 + 500q + 200 dollars when the level of production is q units. The current level of production is 4 units, and the manufacturer is planning to increase this to 4.1 units. Estimate how the total cost will change as a result. Q3 Use a matrix method to find the equilibrium prices and quantities where the supply and demand functions for Good 1, Good 2 and Good 3 are as Qd1 = 50 − 2P1 + 5P2 − 3P3, Qs1 = 8P1 − 5 Qd2 = 22 + 7P1 − 2P2 + 5P3, Qs2 = 12P2 − 5 Qd3 = 17 + P1 + 5P2 − 3P3, Qs3 = 4P3 − 1 Q4 (i) A business manager determines that t months after production begins on a new product, the number of units produced will be P thousand, where P(t) = 6?2 + 5? (? + 1) 2. What happens to production in the long run? (ii) A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture. Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0), but for x > 0, it is found that y = 0.5(x2 + 3x) x3 + x2 + 4x. Assuming the oil slick is continuously distributed, how thick would you expect it to be at the source? Q5 (i) Find the rate of change of the function f(x) = x + 2 1 − 8x with respect to x when x = 1. (ii) The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled by Q(K) = 500 K2/3. Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, where K(t) = 2t4 + 3t + 149 t + 2 (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production by increasing or decreasing at this time?
Q1
It costs a baker a fixed cost of $420 and a variable cost of $2.10 per cupcake. A cupcake is sold for $4.90 each. (i) Make a table showing the cost of producing 20,40,60,80 and 100 cupcakes.
(ii) Make a table showing the revenue from selling 20,40,60,80 and 100 cupcakes.
(iii) Write an algebraic expression representing the cost C as a function of the number of cupcakes x that are produced.
(iv) Write an algebraic expression representing the revenue R as a function of the number of cupcakes x sold.
(v) Graph both functions on the same coordinate axes.
(vi) From your graph find coordinate at which cost equals revenue.
(vii) Using your graph, determine how many cupcakes need to be made to produce revenue of at least $1,029.How much profit is made for this number of cupcakes?
? ?.
(i) Discuss the continuity of the function f(x) = {x2 − 3x if x < 2 2x + 4 if x ≥ 2
on the open interval 0 < x < 2 and on the closed interval 0 ≤ x ≤ 2.
(ii) An efficiency study of the morning shift at a certain factory indicates that an average worker arriving on the job at 7:00 A.M.will have assembled f (x) = x3 + 6x2 + 15x units x hours later. Approximately how many units will the worker assemble between 8:00 and 8:15 A.M.?
(iii) A manufacturer’s total cost is C(q) = 0.1q3 + 0.5q2 + 500q + 200 dollars when the level of production is q units. The current level of production is 4 units, and the manufacturer is planning to increase this to 4.1 units. Estimate how the total cost will change as a result.
Q3
Use a matrix method to find the equilibrium prices and quantities where the supply and demand functions
for Good 1, Good 2 and Good 3 are as
Qd1 = 50 − 2P1 + 5P2 − 3P3, Qs1 = 8P1 − 5
Qd2 = 22 + 7P1 − 2P2 + 5P3, Qs2 = 12P2 − 5
Qd3 = 17 + P1 + 5P2 − 3P3, Qs3 = 4P3 − 1
Q4
(i) A business manager determines that t months after production begins on a new product,
the number of units produced will be P thousand, where P(t) = 6?2 + 5? (? + 1) 2.
What happens to production in the long run?
(ii) A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture. Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0), but for x > 0, it is found that
y = 0.5(x2 + 3x) x3 + x2 + 4x.
Assuming the oil slick is continuously distributed, how thick would you expect it to be at the source?
Q5
(i) Find the rate of change of the function f(x) = x + 2 1 − 8x with respect to x when x = 1.
(ii) The number of units Q of a particular commodity that will be produced with K
thousand dollars of capital expenditure is modeled by Q(K) = 500 K2/3.
Suppose that capital expenditure varies with time in such a way that t months from now
there will be K(t) thousand dollars of capital expenditure, where
K(t) = 2t4 + 3t + 149 t + 2
(a) What will be the capital expenditure 3 months from now? How many units will be produced at this time?
(b) At what rate will production be changing with respect to time 5 months from now?
Will production by increasing or decreasing at this time?
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