Mathematics Questions and Answers

Let \(x\) = number of VIP tickets sold and

\(y\) = number of regular tickets sold

Total number of tickets sold = 200

⇒ \(x\) + \(y\) = 200 ---- (i)

If it costs ₦1,200 for a VIP ticket, then it costs ₦1200x for \(x\) number of VIP tickets sold and

If it costs ₦700 for a regular ticket, then it costs ₦700\(y\) for \(y\) number of VIP tickets sold

The total amount realised from the sale of tickets = ₦180,000

⇒ 1200\(x\) + 700\(y\) = 180000 ----- (ii)

From equation (i)

\(x\) = 200 - \(y\) ----- (iii)

Substitute (200 - \(y\)) for \(x\) in equation (ii)

⇒ 1200(200 - \(y\)) + 700\(y\) = 180000

⇒ 240000 - 1200\(y\) + 700\(y\) = 180000

⇒ 240000 - 500\(y\) = 180000

Collect like terms

⇒ 240000 - 180000 = 500\(y\)

⇒ 60000 = 500\(y\)

⇒ \(y = \frac{60000}{500} = 120\)

Substitute 120 for \(y\) in equation (iii)

⇒ \(x = 200 - 120\)

⇒ \(x = 80\)

∴ The total number of VIP tickets sold is 80 and regular is 120

The two trailing zeros in the number are not significant, but the other five are, making it a five-figure number.

|AP| = |PB| = \(x\) (The perpendicular to a chord bisects the chord if drawn from the center of the circle.)

From ∆OPB

Using Pythagoras theorem

⇒ \(13^2 = 12^2 + x^2\)

⇒ \(169 = 144 + x^2\)

⇒ \(169 - 144 = x^2\)

⇒ \(x^2 = 25\)

⇒ \(x = \sqrt25 = 5 cm\)

∴ Length of the chord |AB| = \(x  + x  = 5 + 5 = 10 cm\)

On the \(y\)-axis, each box is \(\frac{1 - 0}{5} = \frac{1}{5}\) = 0.2unit

On the \(x\)-axis, each box is \(\frac{90 - 0}{6} = \frac{90}{6} = 15^o\)

⇒ \(θ_1 = 180^o + (2.5\times15^o) = 180^o + 37.5^o = 217.5^o ≃ 218^o \)(2 and half boxes were counted to the right of 180\(^o\))

⇒ \(θ_2 = 270^o + (3.5\times15^o) = 270^o + 52.5^o = 322.5^o ≃ 323^o \)(3 and half boxes were counted to the right of 270\(^o\))

∴ \(θ = 218^o, 323^o\)

AB = \(\frac{θ}{360}\times 2\pi Rcos\propto\) (distance on small circle)

= 64 - 56 = 8\(^o\)

\(\propto = 86^o\)

⇒ AB = \(\frac{8}{360}\) x 2 x 3.142 x 6370 x cos 86

⇒ AB = \(\frac{22,338.29974}{360}\)

∴ AB = 62km (to the nearest km)