1)The position vectors of R,S and T are 2j-j,8i+11j and 12i+19j respectively.Show that R,S and T lie on a straight line.
2)From the third floor of a building,an engineer observes a water tower,50 metres away from the building as follows:the angle of depression of the foot of the tower is 19° while the angle of elevation of the top of the tower is 28°.Calculate the height of the tower,correct to one decimal place.
3)The capacity of a giant model of a bottle is 4050 litres.The actual bottle has a volume 1200cm^3 and a height of 18cm.Calculate the height of the model bottle in metres.
4)Three partners;Awino,Braxton and Cate contributed Kshs 750000 ,Kshs 600000 and Kshs 900000 respectively to start a business of a minibus plying Kisumu-Bondo route.The minibus carries 33 passengers each paying Kshs 300.The minibus makes two round trips each day and always full.Each day Kshs 10500 is used to cover operational cost and wages.
a)Calculate their net profit per day.
b)The matatu works for 25 days per month and is served every month at a cost of Kshs 15000.Calculate their monthly profit for the month of September.
c)The three partners agreed to save 40% of the profit,share 24% equally and the rest in the ratio of their contribution.Calculate Braxton's share in the month of November.
d)The minibus developed mechanical problems and they decided to sell it through a dealer who charged a commission of 4% on selling price.Each partner received Kshs 625000 from the dealer after he had taken his commission .Determine the price at which the dealer sold the minibus.
5)a)The figure below shows a velocity-time graph of a particle starts from rest and accelerates at 11/4metres per second ^2 for t1,seconds until it attains a velocity of 22m/sIt then maintains this velocity upto the 40th second.Breaks are applied bringing it uniformly to rest after t2.The particle covers 847 metres before it comes to rest.
Find;
i)The value of t1
ii)The deceleration time
b)The distance between two trains T1 and T2 travelling towards the same direction is 100metres.T1 is 60 metres long and travels at 80k/h while T2 is 140 metres long and travels at 60km/h.Find the time in seconds it takes T1 to completely overtake T2.
1) To show that R, S, and T lie on a straight line, we need to show that the vectors from R to S and from S to T are collinear.
The vector from R to S is given by S - R = (8i + 11j) - (0i + 2j) = 8i + 9j = 9(8/9i + 1j).
The vector from S to T is given by T - S = (12i + 19j) - (8i + 11j) = 4i + 8j = 4(1i + 2j).
We can see that the vectors from R to S and from S to T are multiples of each other, which means they are collinear. Therefore, R, S, and T lie on a straight line.
2) Let h be the height of the water tower.
From the top of the building, the engineer's line of sight to the top of the tower forms an angle of elevation of 28°. This means that the height of the tower is h = 50tan(28°).
From the third floor of the building, the engineer's line of sight to the foot of the tower forms an angle of depression of 19°. This means that the height of the tower is h/50 = tan(19°).
Equating the two expressions for h gives 50tan(28°) = tan(19°), and solving for h gives h = 50tan(28°)/tan(19°). Calculating this using a calculator gives h ≈ 47.5 meters.
3) The volume of the model bottle is 4050 liters, or 4050 * 1000 cm^3.
The volume of the actual bottle is 1200 cm^3.
Let h be the height of the model bottle.
Since the volumes of the model bottle and the actual bottle are proportional to the cubes of their heights, we can set up the equation (h/18)^3 = 4050/1200.
Simplifying this equation gives h^3 = 4050/1200 * 18^3.
Taking the cube root of both sides gives h = (4050/1200 * 18^3)^(1/3). Calculating this gives h ≈ 35.5 meters.
4) a) The total contribution of the partners is 750000 + 600000 + 900000 = 2250000 Kshs.
The income from the minibus per day is 33 passengers * 300 Kshs/passenger = 9900 Kshs.
The net profit per day is the income minus the operational cost and wages, which is 9900 Kshs - 10500 Kshs = -600 Kshs.
b) There are 25 days in September.
The monthly profit is the net profit per day multiplied by the number of working days in a month, which is -600 Kshs * 25 days = -15000 Kshs.
c) The total profit saved is 40% of the profit, which is 0.4 * (-15000 Kshs) = -6000 Kshs.
The amount to be shared equally is 24% of the profit, which is 0.24 * (-15000 Kshs) = -3600 Kshs.
The remaining profit to be shared in the ratio of their contributions is -15000 Kshs - (-6000 Kshs) - (-3600 Kshs) = -5400 Kshs.
The ratio of their contributions is 750000:600000:900000 = 125:100:150.
Braxton's share is 150/375 * (-5400 Kshs) = -2160 Kshs.
d) Each partner received 625000 Kshs after the dealer took his commission of 4%.
Let x be the selling price of the minibus.
625000 Kshs = x - 0.04x = 0.96x.
Solving for x gives x = 625000 Kshs / 0.96 = 651041.67 Kshs.