a. A textbook company discovered that the profit made from selling its books is given by y = \(\frac{x^2}{8}\) + 5x, where x is the number of textbooks sold (in thousands) and y is the corresponding profit (in Ghana Cedis). If the company made a profit of GH₵ 20,000.00
i. form a quadratic equation in x;
ii. (using the quadratic formula, find, correct to the nearest whole number, the number of textbooks sold to make the profit.
b. The angle of elevation of the top T of a tree from a point P on the same ground level as the foot Q of a tree is 28\(^o\). A bird perched at a point R, halfway up the tree.
i. Represent the information in a diagram.
ii. Calculate, correct to the nearest degree, the angle of elevation of R from P.
(a) They were expected to obtain 20,000 = \(\frac{x^2}{8} + 5x\) and thereafter multiply through by 8 and simplify to get \(x ^2\) + 40x - 16,000. Using the quadratic formula
x = \(\frac {-b \pm \sqrt{b^2 - 4ac} }{2a}\)
where a = 1, b = 40 and c = - 160,000
x = \(\frac{-40 \pm \sqrt{(40)^2 - 4(1) (-160000)}} {2(1)}\)
= \(\frac{-40 \pm \sqrt{(40)^2 - 4(1) (-160000)}} {2}\)
= \(\frac{-40 \pm \sqrt{1600 + 640000}}{2}\)
= 38.4997 or -420.4997
The number of books required to make the profit approximately 380,500, correct to the nearest whole number.
(b) The required diagram is
From triangle PQR, tan x = \(\frac{y}{2}\)W
y = 2w tan x..........(1)
Also, from \(\bigtriangleup\) PTQ,
tan 28\(^o\) = \(\frac{y}{w}\)
y = w tan 28\(^o\).........(2)
Equating (1) and (2) yielded 2w tan X = w tan 28\(^o\)
so that tanX = \(\frac{\text{w} tan 28^o}{2w}\) = \(\frac{0.5317}{2}\) which implies that tan X = 0.2659
X = tan\(^{-1}\)(0.2659) = 15.42\(^o\).
The angle of elevation of R from P = 15\(^o\), correct to the nearest degree.
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