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.
Explanation
(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.