9). Hooke’s Law. 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 29 cm when the weight is 4 kg what is the distance when the weight is 9 kg?

10). Find the variation constant and an equation of variation where y varies inversely as x and y =9 when x= 17?
11). It takes 8hr for 9 cooks to prepare the food for a wedding rehearsal dinner. How long will it take 3 cook to prepare the dinner? Also what variation applies Inverse or Direct?
12). The stopping distance d of a car after the brakes are applied varies directly as the square of the speed r. If a car traveling 80 mph can stop in 360ft, how many feet will it take the same car to stop at a speed of 70?

9 find k. k=4kg/29cm

distance=4/29 * 9 I am very disturbed by the questions statement that kilograms is weight. It is not weight, it is mass.
10 y=k/x put in x,y and solve for k.

11 time=cook/rate=3/(9/8hr)

9) To find the distance when the weight is 9 kg, we can use Hooke's Law, which states that the distance varies directly as the weight. This can be expressed as:

d ∝ w

Given that the distance is 29 cm when the weight is 4 kg, we can set up a proportion to solve for the unknown distance (x) when the weight is 9 kg:

29 cm / 4 kg = x cm / 9 kg

Cross-multiplying, we have:

4 * x = 29 * 9

4x = 261

Dividing both sides by 4, we get:

x = 261 / 4

x = 65.25 cm

Therefore, the distance when the weight is 9 kg is 65.25 cm.

10) When y varies inversely as x, it means that the product of y and x remains constant. We can represent this as:

y * x = k

where k is the variation constant.

Given that y = 9 when x = 17, we can substitute these values into the equation to solve for the variation constant:

9 * 17 = k

153 = k

Therefore, the variation constant is 153 and the equation of variation is:

y = 153 / x

11) In this problem, it takes 8 hours for 9 cooks to prepare the food. We want to find how long it will take 3 cooks to prepare the dinner.

If the time required to prepare the food varies inversely with the number of cooks, we can set up a proportion to solve for the unknown time (x):

8 hr / 9 cooks = x hr / 3 cooks

Cross-multiplying, we have:

8 * 3 = 9 * x

24 = 9x

Dividing both sides by 9, we get:

x = 24 / 9

x = 2.67 hr

Therefore, it will take approximately 2.67 hours for 3 cooks to prepare the dinner. The variation in this case is inverse.

12) In this problem, the stopping distance (d) of a car after the brakes are applied varies directly as the square of the speed (r). We are given that a car traveling at 80 mph can stop in 360 ft.

Using the direct variation equation:

d = k * r^2

we can substitute the values given (d = 360 ft and r = 80 mph) to solve for the variation constant (k):

360 = k * (80^2)

360 = k * 6400

Dividing both sides by 6400, we get:

k = 360 / 6400

k = 0.05625

Therefore, the variation constant is 0.05625.

To find the distance it will take the same car to stop at a speed of 70 mph, we can substitute this value into the variation equation:

d = 0.05625 * (70^2)

d = 274.3125 ft

Therefore, it will take approximately 274.3125 feet for the same car to stop at a speed of 70 mph.

9) To solve this problem, we can use Hooke's Law and set up a proportion. Hooke's Law states that the distance, d, varies directly with the weight, w. This relationship can be written as: d = kw, where k is the constant of variation.

Given that d = 29 cm when w = 4 kg, we can substitute these values into the equation to find the value of k.

29 = k * 4

To find k, divide both sides of the equation by 4:

k = 29 / 4

Now that we have the value of k, we can use it to find the distance, d, when the weight, w, is 9 kg. Substitute the new weight into the equation:

d = k * w
d = (29 / 4) * 9

Simplify the expression:

d = 261 / 4

Therefore, when the weight is 9 kg, the distance is 261/4 cm.

10) To find the variation constant and equation of variation, we are given that y varies inversely with x. This relationship can be written as: y = k/x, where k is the constant of variation.

Given that y = 9 when x = 17, we can substitute these values into the equation to find the value of k.

9 = k / 17

To find k, multiply both sides of the equation by 17:

k = 9 * 17

Therefore, the variation constant is k = 153.

Now we can write the equation of variation:

y = 153 / x

11) To find how long it will take 3 cooks to prepare the dinner given that it takes 8 hours for 9 cooks, we can use the concept of direct variation. Direct variation states that if one quantity is multiplied by a constant factor, then the other quantity is multiplied by the same factor.

In this case, the number of cooks is inversely proportional to the time taken. So, if the number of cooks decreases to 3 from 9, the time taken will increase by the same factor.

To find the time it will take 3 cooks to prepare the dinner, we set up a proportion:

8 / 9 = x / 3

Cross-multiply and solve for x:

9x = 3 * 8
9x = 24
x = 24 / 9

Therefore, it will take 3 cooks approximately 2.67 hours to prepare the dinner.

12) To find the stopping distance of a car at a speed of 70 mph, we are given that the stopping distance, d, varies directly with the square of the speed, r. This can be written as: d = kr^2, where k is the constant of variation.

Given that the car can stop in 360 ft when traveling at 80 mph, we can substitute these values into the equation to find the value of k.

360 = k * (80^2)

To find k, divide both sides of the equation by (80^2):

k = 360 / (80^2)

Now we can use k to find the stopping distance, d, when the speed, r, is 70 mph:

d = k * (70^2)
d = k * 4900

Substitute the value of k we found earlier:

d = (360 / (80^2)) * 4900

Simplify the expression:

d = 1965 ft

Therefore, it will take approximately 1965 ft for the car to stop at a speed of 70 mph.