Mathematics Questions and Answers

First, we find the equation of the boundary line using the two intercepts.The slope is

m = $\frac{4 - 0}{0 - 5} = {4}{5}$

The y-intercept is 4

The slope-intercept form of the equation is therefore

y = -$\frac{4}{5} x + 4$

$\implies y + \frac{4}{5} x = 4$

Multiply both sides by $\frac{5}{4}$

$\implies\frac{5}{4}(y +\frac{4}{5} x) = 4\times\frac{5}{4}$

$\implies\frac{5}{4} y + x = 5$

Multiply both sides by 4

⇒ $5y + 4x = 20$

The inequality is therefore either $5y + 4x ≤ 20$ or $5y + 4x ≥ 20.$

Using the test point (0, 0) -The origin

⇒ 5(0) + 4(0) ≤ 20

⇒ 0 ≤ 20 (True)

∴ The inequality is $5y + 4x ≤ 20$

Intersecting Chords Theorem states that If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal.

⇒ AE * EB = CE * ED

⇒ 6 * $x$ = 4 * ($x$ + 5)

⇒ 6$x$ = 4$x$ + 20

⇒ 6$x$ - 4$x$ = 20

⇒ 2$x$ = 20

∴ $x = \frac{20}{2}$ = 10 units

Area of the composite figure = Area of semi circle + Area of rectangle + Area of triangle

Area of semi circle = $\frac{1}{2}\pi r^2 = \frac{1}{2}\times\pi\times\frac{d^2}{4} = \frac{1}{2}\times\frac{22}{7}\times\frac{42^2}{4} = 693 m^2$

Area of rectangle = l x b = 42 x 60  =2520 m$^2$

Area of triangle = $\frac{1}{2}\times b \times h = \frac{1}{2}\times 36 \times 42 = 756 m^2$

∴ Area of the composite figure = 693 + 2520 + 756 = 3969 m$^2$

$log_2 (6 - x) = 3 - log_2 x$

⇒ $log_2 (6 - x) = 3 log_2 2 - log_2 x$ (since $log_2$ 2 = 1)

⇒ $log_2 (6 - x) = log_2 2^3 - log_2 x$ $(a log$ c = $log$ c$^a)$

⇒ $log_2 (6 - x) = log_2 8 - log_2 x$

⇒$log_2 (6 - x) = log_2 \frac{8}{x}$ ($log$ a - $log$ b = $log \frac{a}{b})$

⇒ $6 - x = \frac{8}{x}$

⇒ $x (6 - x) = 8$

⇒ $6x - x^2 = 8$

⇒ $x^2 - 6x + 8 = 0$

⇒ $x^2 - 4x - 2x + 8 = 0$

⇒ $x (x - 4) - 2(x - 4) = 0$

⇒ $(x - 4)(x - 2) = 0$

⇒ $x - 4 = 0 or x - 2 = 0$

∴ x = 4 or 2

Let number of children's ticket at ₦250.00 each = $x$

∴ Number of adult tickets at ₦520.00 each = 5$x$

Then,

Total amount of money received from children's tickets = 250$x$

Total amount of money received from adult tickets = 520(5$x$)

⇒ 250$x$ + 520(5$x$) = 171,000

⇒ 250$x$ + 2600$x$ = 171,000

⇒ 2850$x$ = 171,000

⇒ $x = \frac{171,000}{2850} = 60$

∴ 60 tickets were sold at ₦250.00 and 300 tickets were sold at ₦520.00