1-)A passenger boat which travels 6 mph faster than a freighter required 3 days less time to make a trip of 2592 miles.
Find the rate of the passenger boat.
2-)For the first of two basketball games the price of each reserved seat was $1.50,the price of each unreserved seat was 5 cents, and the number of reserved seats sold was 1/25 of the number of unreserved seats sold.If, at a second game, the price of the reserved seats is to be reduced to $1.00,what proportion should be reserved to give the same receipts, provided the same number of seats are sold?
THANK YOU FOR THE HELP !!
1-) To find the rate of the passenger boat, let's break down the information given in the problem:
- The passenger boat travels 6 mph faster than the freighter.
- The passenger boat takes 3 days less time than the freighter to make a trip of 2592 miles.
Let's denote the rate of the freighter as "x" mph. Therefore, the rate of the passenger boat would be "x + 6" mph.
We know that time equals distance divided by rate. So, the time it takes for the freighter to travel 2592 miles would be "2592 / x" hours, and the time it takes for the passenger boat would be "2592 / (x + 6)" hours.
According to the problem, the passenger boat takes 3 days less time than the freighter. Since there are 24 hours in a day, this can be represented as:
2592 / x - (2592 / (x + 6)) = 3 * 24 (since 3 days less = 3 * 24 hours)
To solve this equation, we can multiply through by the least common denominator (x(x + 6)):
2592(x + 6) - 2592x = 3 * 24 * x(x + 6)
After simplifying the equation, we get:
2592 * 6 = 3 * 24 * x^2 + 3 * 24 * 6x
Simplifying further:
15552 = 72x^2 + 432x
Rearranging the equation to the standard quadratic form:
72x^2 + 432x - 15552 = 0
To solve this quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values from our equation above:
x = (-432 ± √(432^2 - 4 * 72 * -15552)) / (2 * 72)
By solving this equation, you will find two possible values for x. Choose the positive value, as we're looking for the rate of the freighter, which cannot be negative. This value will be the rate of the freighter. Then, add 6 to find the rate of the passenger boat.
2-) In order to determine the proportion of reserved seats that should be sold to achieve the same receipts, we can break down the given information:
- For the first game, reserved seats were sold for $1.50 and unreserved seats for $0.05.
- The number of reserved seats sold was 1/25th of the number of unreserved seats sold.
Let's denote the number of unreserved seats sold as "x". Therefore, the number of reserved seats sold would be (1/25)x.
Now, we need to calculate the total receipts from the first game. The total receipts are equal to the sum of the revenue from reserved seats and the revenue from unreserved seats.
For reserved seats, the revenue is the price per seat multiplied by the number of seats sold. Therefore, the revenue from reserved seats would be:
1.50 * (1/25)x
For unreserved seats, the revenue is also the price per seat multiplied by the number of seats sold. Therefore, the revenue from unreserved seats would be:
0.05 * x
The total receipts would be the sum of these two revenues:
1.50 * (1/25)x + 0.05x
Now, we need to determine the proportion of reserved seats that should be sold in the second game to achieve the same receipts. Let's denote this proportion as "p", where 0 <= p <= 1.
For the second game, the price of reserved seats is reduced to $1.00. Therefore, the revenue from reserved seats in the second game would be:
1.00 * p * x
The revenue from unreserved seats in the second game would remain the same:
0.05 * x
The total receipts from the second game would be the sum of these two revenues:
1.00 * p * x + 0.05 * x
Since we want the total receipts from the second game to be the same as the total receipts from the first game, we set these two expressions equal to each other:
1.50 * (1/25)x + 0.05x = 1.00 * p * x + 0.05 * x
To solve for "p", we can simplify the equation:
0.06x = 0.95px
Dividing both sides by "x", we get:
0.06 = 0.95p
Finally, solving for "p" by dividing both sides by 0.95:
p = 0.06 / 0.95