(b) A tank holds 2500 litres of oil.
A small pump can add oil to the tank at a rate of x litres per minute.
A large pump can add oil to the tank at a rate of (x + 20) litres per minute.
(i) Write down an expression, in terms of x, for the number of minutes the small pump takes to fill the empty tank.
Answer ........................................... [1]
(ii) Write down an expression, in terms of x, for the number of minutes the large
pump takes to fill the empty tank.
(iii) It takes 15 minutes longer to fill the empty tank using the small pump than it does with the large pump.
Form an equation in x and show that it simplifies to 3x2 + 60x - 10 000 = 0.
Answer ........................................... [1]
[4]
(iv) Solve the equation 3x2 + 60x – 10 000 = 0. Give each answer correct to 2 decimal places.
2
Answer x = ...................... or ...................... [4] (v) Find the length of time the large pump takes to fill the empty tank.
Give your answer in minutes and seconds, correct to the nearest second.
Answer .............. minutes .............. seconds [3]
Answers
Answer:
hi to everyone who has a lot of central Florida and Alabama
Given:
A tank holds 2500 liters of oil.
A small pump can add oil to the tank at a rate of x liters per minute.
A large pump can add oil to the tank at a rate of (x + 20) liters per minute.
To Find:
(i). Write down an expression, in terms of x, for the number of minutes the small pump takes to fill the empty tank.
(ii). Write down an expression, in terms of x, for the number of minutes the large pump takes to fill the empty tank.
(iii). It takes 15 minutes longer to fill the empty tank using the small pump than it does with the large pump. Form an equation in x and show that it simplifies to 3x2 + 60x - 10 000 = 0.
(iv). Solve the equation 3x2 + 60x – 10 000 = 0. Give each answer correct to 2 decimal places.
(v). Find the length of time the large pump takes to fill the empty tank. Give your answer in minutes and seconds, correct to the nearest second.
Solution:
(i). According to the question,
volume = 2500 liters.
rate for small pump = x liters/min.
So, Time = Volume / rate.
Time = (2500 / x) min.
Hence, the expression, in terms of x, for the number of minutes the small pump takes to fill the empty tank is (2500 / x) min.
(ii). According to the question,
volume = 2500 liters.
rate for large pump = (x + 20)min.
So, Time = Volume / rate.
Time = 2500 / (x + 20) min.
Hence, the expression, in terms of x, for the number of minutes the large
pump takes to fill the empty tank is 2500 / (x+20) min.
(iii). Time for small pump = (2500 / x) min.
Time for the large pump = 2500 / (x + 20) min.
If it takes 15 minutes for small pump to fill the tank,
The equation will be,
(2500 / x) + 15 = 2500 / (x+20).
(2500 + 15x) / x = 2500 / (x+20).
(2500 + 15x)(x + 20) = 2500x.
2500x + 50000 + 15x² + 300x = 2500x.
15x² + 300x + 50000 = 0.
5 (3x² + 60x + 10000) = 0.
3x² + 60x + 10000 = 0.
Hence, the equation in x is 2500 / x) + 15 = 2500 / (x+20) and when we solve it, it simplifies to equation 3x2 + 60x - 10000 = 0.
(iv). By the formula,
x = -b ± √(b² - 4ac) / 2a.
So,
x = -60 ± √60² - 4×3×(-10000) / 2×3.
x = -60 ± √3600 + 120000 / 6.
For +,
x = -60 + √123600 /6.
x = 291.567 /6.
x = 48.59.
For -,
x = -60 - √123600 /6.
x = -68.59.
Hence, the values of x in + and - are 48.59 and -68.59.
(v). We already know that,
For large pump, Time = 2500 /(x+20).
So, putting the value of x = 48.59,
Time = 2500 / (48.59 + 20).
Time = 36 minutes 44 seconds.
Hence, the length of time the large pump takes to fill the empty tank is 36 minutes 44 seconds.