Question

It takes a car 4 hours to reach a destination traveling at a speed of 63 km/hr. How long will the way back take if the car travels at a speed of 56 km/hr? Do these quantities (time and speed) vary directly or inversely?
Answer
What is the constant of variation?
Answer
Two small planes leave Sacramento and LA, which are 380 miles apart, and fly towards each other. If one plane flies 30 mph faster than the other, and they meet in an hour, at which speed does each fly?
Answer
Try again
Answer
Five gallons of food weigh 37 pounds. How much does three gallons of food weigh? Find the constant of variation. Do these quantities - weight and volume - vary directly or inversely?
Answered by GPT 3.5
To find the time it takes for the car to return, we can use the formula:
Time = Distance/Speed

Since the distance remains the same for the round trip, we can use the same distance for both calculations.

In the given case, it took 4 hours to reach the destination at a speed of 63 km/hr. So, the distance traveled is:
Distance = Speed * Time = 63 km/hr * 4 hours = 252 km

Now, we can calculate the time it takes for the car to return at a speed of 56 km/hr:
Time = Distance/Speed = 252 km / 56 km/hr ≈ 4.5 hours

The time it takes for the car to return is approximately 4.5 hours.

These quantities (time and speed) vary inversely. When the speed decreases, the time it takes to travel a fixed distance increases.
Answered by GPT 3.5
In direct variation, the constant of variation (often represented by k) is the ratio between the two variables. However, in this problem, we are dealing with inverse variation.

In inverse variation, the constant of variation (often represented by k) is the product of the two variables. The product of speed and time is always constant in inverse variation scenarios.

For this problem, the constant of variation is:

Constant of Variation = Speed * Time = k

So, in this case, the constant of variation is 63 km/hr * 4 hours = 252 km.
Answered by GPT 3.5
Let's assume the speed of the slower plane is x mph.

According to the given information, the speed of the faster plane is 30 mph faster than the slower plane. Therefore, the speed of the faster plane is (x + 30) mph.

We know that distance = speed * time. In this case, the total distance is 380 miles and the time taken for the planes to meet is 1 hour.

For the slower plane:
Distance = speed * time
Distance = x mph * 1 hour
380 = x

For the faster plane:
Distance = speed * time
Distance = (x + 30) mph * 1 hour
380 = x + 30

Now, we can solve the above equations to find the speed of each plane:

From the first equation: x = 380

Substituting the value of x in the second equation:
380 + 30 = x + 30
410 = x + 30
380 = x

So, the speed of the slower plane is 380 mph, and the speed of the faster plane is (380 + 30) mph = 410 mph.
Answered by GPT 3.5
Apologies for the confusion.

Let's assume the speed of the slower plane is x mph.

According to the given information, the speed of the faster plane is 30 mph faster than the slower plane. Therefore, the speed of the faster plane is (x + 30) mph.

When the slower and faster planes start flying towards each other, their combined speed is the sum of their individual speeds:
Combined Speed = Speed of the slower plane + Speed of the faster plane.

The combined speed is determined by the total distance traveled (380 miles) divided by the time taken (1 hour):
Combined Speed = 380 miles / 1 hour.

Therefore, we can write the equation:

Combined Speed = x mph + (x + 30) mph
380 miles / 1 hour.

Simplifying the equation:

2x + 30 = 380 miles / 1 hour.

Now, we can solve for x:

2x + 30 = 380
2x = 380 - 30
2x = 350
x = 350 / 2
x = 175 mph.

So, the speed of the slower plane is 175 mph, and the speed of the faster plane is (175 + 30) mph = 205 mph.
Answered by GPT 3.5
To find the weight of three gallons of food, we can use the constant of variation.

Let's assume the weight of one gallon of food is x pounds.

According to the given information, five gallons of food weigh 37 pounds. Therefore, we can set up the following proportion:

5 gallons / 37 pounds = 3 gallons / (weight of 3 gallons)

Simplifying this proportion:

(5/37) = (3/x)

To find the weight of three gallons, we can cross-multiply and solve for x:

5x = 37 * 3
5x = 111
x = 111 / 5
x = 22.2 pounds

So, three gallons of food weigh 22.2 pounds.

Now, let's determine the constant of variation. In this case, it is the ratio of weight to volume. We have:

Constant of Variation = Weight / Volume = k

Using the initial information, we can substitute the values and solve for k:

k = 37 pounds / 5 gallons

k = 7.4 pounds/gallon

The constant of variation is 7.4 pounds/gallon.

These quantities - weight and volume - vary directly. As the volume of food increases, the weight also increases proportionally.