please can somebody explain to me how to do this questions
8. Elli travelled 700km by truck to deliver some ostriches to her friend's farm. she then returned the same distance by car. The car averaged 10km/h faster than the truck and took 1 hour less time. What was the speed of the car?
9. When two pumps are turned on they can fill the pond at Cassandra's farm in seven hours. individually, one pump takes two hours longer than the other to fill the pond. Find the time to fill the pond by the faster pump by itself.
10. A boat travelled 30km upstream, against the current, to deliver some feed from Elli, to Helmut's farm, it then turned around and headed 35km downstream to dock for the night. If the speed of the current was 3km/h and the upstream trip took 45 mins longer than the downstream trip, what was the speed of the boat?
this unit is about solving quadratics with the quadratic formula
If the car's speed was s, then the truck's speed was s-10.
Since time = distance/speed, and the truck took an extra hour,
700/(s-10) = 700/s + 1
clearing the fractions, you get
700s = 700(s-10) + 1(s)(s-10)
s^2 - 10s - 7000 = 0
Now, using the quadratic formula, you have
s = (10±√(100+28000))/2 = 5±√7025 = 5(1±√281)
You want the positive value.
For the pumps, you have the usual work problem, so you have
1/x + 1/(x+2) = 1/7
#10 is pretty much like #8. How far do you get on it?
Sure, I'd be happy to help explain how to solve these questions!
8. To find the speed of the car, we can set up a system of equations based on the given information. Let's say the speed of the truck is "x" km/h. Since the car averaged 10 km/h faster, the speed of the car would be "x + 10" km/h.
To find the time it takes for each vehicle to complete the trip, we can use the formula: time = distance / speed.
For the truck, the time taken would be 700km / x km/h, and for the car, it would be 700km / (x + 10) km/h.
Now, we know that the car took 1 hour less time than the truck, so we can set up the equation: 700km / (x + 10) = 700km / x - 1.
To solve this equation, we can cross-multiply and simplify:
700km * x = 700km * (x + 10) - x - 10.
Simplifying further, we get:
700x = 700x + 7000 - x - 10.
Combining like terms, we have:
0 = 699x + 6990.
Dividing both sides of the equation by 699, we get:
x = 10.
So, the speed of the car is 10 + 10 = 20 km/h.
9. Let's say the time taken by the faster pump to fill the pond is "x" hours. Then, the slower pump would take x + 2 hours.
We can set up another equation based on the given information that when both pumps work together, they can fill the pond in 7 hours. Using the formula: work rate = 1/time, the equation becomes:
1/x + 1/(x + 2) = 1/7.
To solve this equation, we can cross-multiply and simplify:
7(x + 2) + 7x = x(x + 2).
Expanding and simplifying, we get:
7x + 14 + 7x = x^2 + 2x.
Rearranging the terms, we have:
x^2 - 12x - 14 = 0.
Now, we can solve this quadratic equation using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a).
For our equation, a = 1, b = -12, and c = -14. Plugging these values into the quadratic formula, we can calculate the two possible solutions for x.
Once we find the values of x, we can determine the time taken by the faster pump by itself.
10. To solve this question, we can set up another system of equations based on the given information. Let's say the speed of the boat is "b" km/h.
Since the speed of the current is given as 3 km/h, the effective speed of the boat upstream will be reduced by 3 km/h, and downstream it will be increased by 3 km/h. So, the speed of the boat upstream would be "b - 3" km/h, and downstream it would be "b + 3" km/h.
We are also given that the upstream trip took 45 minutes longer than the downstream trip. Since 45 minutes is equivalent to 0.75 hours, we can set up the equation:
30km / (b - 3) = 35km / (b + 3) + 0.75.
To solve this equation, we can cross-multiply and simplify:
35(b - 3) + 0.75(b + 3) = 30(b + 3).
Expanding and simplifying, we get:
35b - 105 + 0.75b + 2.25 = 30b + 90.
Combining like terms, we have:
35b + 0.75b - 30b = 105 - 2.25 - 90.
Simplifying further, we get:
5.75b = 12.75.
Dividing both sides of the equation by 5.75, we find:
b ≈ 2.217 km/h.
Therefore, the speed of the boat is approximately 2.217 km/h.
I hope this explanation helps you understand how to solve these questions!