(a) (i) Illustrate the following statements in a Venn diagram : All good Literature students in a school are in the General Arts class.
(ii) Use ths diagram to determine whether or not the following are valid conclusions from the given statement.
(1) Vivian is in the General Arts class therefore she is a good Literature student.
(2) Audu is not a good Literature student therefore he is not in the General Arts class;
(3) Kweku is not in the General Arts class therefore he is not a good Literature student.
(b) The cost (c) of producing n bricks is the sum of a fixed amount, h, and a variable amount, y, where y varies directly as n. If it costs GH¢950.00 to produce 600 bricks and GH¢ 1,030.00 to produce 1000 bricks,
(i) Find the relationship between c, h and n ; (ii) Calculate the cost of producing 500 bricks.
Explanation
(a) (i) Let :
L = { all good literature students in the school}
A = {all students in the General Arts class}
Then the Venn diagram below represents the statement : 'All good Literature students in a school are in the General Arts class'.
where \(\in\) = {all students in the school}.
(ii) (1) Invalid conclusion
(2) Invalid conclusion
(3) Valid conclusion
(b) (i) c = h + y
where \(y \propto n \implies y = kn\)
\(\implies c = h + kn\)
When n = 600 bricks, c = GH¢ 950.00
\(950.00 = h + 600k ....... (1)\)
When n = 1000 bricks, c = GH¢ 1,030.00
\(1,030.00 = h + 1000k ....... (2)\)
Subtracting (1) from (2), we have
\(1,030 - 950 = 1000k - 600k\)
\(80 = 400k\)
\(k = \frac{80}{400} = \frac{1}{5}\)
\(\therefore c = h + \frac{1}{5}n\)
From (1), \(950 = h + 600k\)
\(950 = h + (600 \times \frac{1}{5})\)
\(950 = h + 120\)
\(h = 950 - 120 = 830\)
\(c = 830 + \frac{1}{5}n\)
(ii) Using
\(c = 830 + \frac{1}{5}n\), we have
\(c = 830 + \frac{1}{5} \times 500\)
= \(830 + 100\)
= \(GH¢ 930.00\)
Hence, the cost of producing 500 bricks = GH¢ 930.00.