(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.
(a)(i) Venn diagram: Draw a rectangle for the school. Inside it draw a circle for the General Arts class \((A)\). Entirely within that circle draw a smaller circle for good Literature students \((L)\). This shows \(L \subset A\): every good Literature student is in the General Arts class, but not every General Arts student is a good Literature student.
(ii) Testing the conclusions:
- Vivian is in General Arts, therefore she is a good Literature student. NOT valid: being in \(A\) does not place her inside the smaller circle \(L\).
- Audu is not a good Literature student, therefore he is not in General Arts. NOT valid: he could be in \(A\) but outside \(L\).
- Kweku is not in General Arts, therefore he is not a good Literature student. VALID: since \(L \subset A\), anyone outside \(A\) must be outside \(L\).
(b) Cost \(c = h + y\) where \(y = kn\) (\(y\) varies directly as \(n\)), so \(c = h + kn\).
\(950 = h + 600k \quad(1)\)
\(1030 = h + 1000k \quad(2)\)
\((2) - (1):\ 80 = 400k \ \Rightarrow\ k = 0.2\). Then \(h = 950 - 600(0.2) = 830\).
(i) \(\mathbf{c = h + 0.2n}\), that is \(c = 830 + 0.2n\) (with \(h = \text{GH¢}830\)).
(ii) For \(n = 500\): \(c = 830 + 0.2(500) = 830 + 100 = \mathbf{\text{GH¢}930.00}\).
(a)(i) Venn diagram: Draw a rectangle for the school. Inside it draw a circle for the General Arts class \((A)\). Entirely within that circle draw a smaller circle for good Literature students \((L)\). This shows \(L \subset A\): every good Literature student is in the General Arts class, but not every General Arts student is a good Literature student.
(ii) Testing the conclusions:
- Vivian is in General Arts, therefore she is a good Literature student. NOT valid: being in \(A\) does not place her inside the smaller circle \(L\).
- Audu is not a good Literature student, therefore he is not in General Arts. NOT valid: he could be in \(A\) but outside \(L\).
- Kweku is not in General Arts, therefore he is not a good Literature student. VALID: since \(L \subset A\), anyone outside \(A\) must be outside \(L\).
(b) Cost \(c = h + y\) where \(y = kn\) (\(y\) varies directly as \(n\)), so \(c = h + kn\).
\(950 = h + 600k \quad(1)\)
\(1030 = h + 1000k \quad(2)\)
\((2) - (1):\ 80 = 400k \ \Rightarrow\ k = 0.2\). Then \(h = 950 - 600(0.2) = 830\).
(i) \(\mathbf{c = h + 0.2n}\), that is \(c = 830 + 0.2n\) (with \(h = \text{GH¢}830\)).
(ii) For \(n = 500\): \(c = 830 + 0.2(500) = 830 + 100 = \mathbf{\text{GH¢}930.00}\).
