The distance d when a spring is stretched by a hanging object varies directly as the weight w of the object. If the distance is 34 cm when the weight is 3 kg, what is the distance when the weight is 9 kg?
22. Find the variation constant and an equation of variation where y varies inversely as x and y=1 when x=9.
23. It takes 6 hr for 4 cooks to prepare the food for a wedding rehearsal dinner. How long will it take 2 cooks to prepare the dinner? A. What kind of variation applies to this situation? B. Solve the problem.
For question 1:
The distance, d, when a spring is stretched by a hanging object varies directly as the weight, w, of the object. This means that the ratio of d to w remains constant.
To find the variation constant and the distance when the weight is 9 kg, we can use the given information that the distance is 34 cm when the weight is 3 kg.
1. Start by finding the variation constant (k):
d1 / w1 = d2 / w2 (where d1 = 34 cm, w1 = 3 kg, d2 = unknown, w2 = 9 kg)
Substitute the values:
34 cm / 3 kg = d2 / 9 kg
Solve for d2:
d2 = (34 cm / 3 kg) * 9 kg
d2 = 306 cm
So, the distance when the weight is 9 kg is 306 cm.
For question 2:
The equation of variation where y varies inversely as x can be written as y = k/x, where k is the variation constant. To find the constant and the equation of variation, we can use the given information that y = 1 when x = 9.
1. Substitute the values into the equation:
1 = k / 9
2. Solve for k:
k = 1 * 9
k = 9
3. Write the equation of variation:
y = 9/x
For question 3:
It takes 6 hours for 4 cooks to prepare the food for a wedding rehearsal dinner. We want to find out how long it will take 2 cooks to prepare the dinner.
This situation represents inverse variation, where the number of cooks is inversely proportional to the time taken.
A. Inverse variation applies to this situation because as the number of cooks decreases, the time taken increases.
B. To solve the problem, we can use the formula for inverse variation:
n1 * t1 = n2 * t2 (where n1 = 4 cooks, t1 = 6 hr, n2 = 2 cooks, t2 = unknown)
Substitute the values:
4 cooks * 6 hr = 2 cooks * t2
Solve for t2:
24 hr = 2 cooks * t2
t2 = 24 hr / 2 cooks
t2 = 12 hr
So, it will take 2 cooks 12 hours to prepare the dinner.