A parcel delivery service charges $2.50 to deliver packages that weight 2 lb. The delivery packages weighing over 2 lb, but less than 3 1/2 lb, is $2.95. For packages weighing over 3 1/2 lb, but less than 5 lb, the fee is $3.40, and so on. Rosann uses the service to send a package and is charged $4.75. What is the range of the weight of her package?
Help, I don't know how much each intervals cost!
we have the piecewise cost function for weight w,
c(w) =
2.50 for w <= 2
2.95 for 2 < x <= 3.5
3.40 for 3.5 < x <= 5
...
2.50 + .45⌈(w-2)/1.5⌉
So, now we want w such that
2.50 + .45⌈(w-2)/1.5⌉ = 4.75
.45⌈(w-2)/1.5⌉ = 2.25
⌈(w-2)/1.5⌉ = 5
4 < (w-2)/1.5 <= 5
6 < w-2 <= 7.5
8 < w <= 9.5
check:
continuing our piecewise table, we get
3.40 for 3.5 < x <= 5
3.85 for 5 < w <= 6.5
4.30 for 6.5 < w <= 8
4.75 for 8 < w <= 9.5
To determine the weight intervals and corresponding costs for the parcel delivery service, we can analyze the given information. Let's break it down step by step:
1. We know that the service charges $2.50 for packages weighing exactly 2 lb.
2. We also know that the cost for packages weighing over 2 lb but less than 3 1/2 lb is $2.95.
3. Additionally, the cost for packages weighing over 3 1/2 lb but less than 5 lb is $3.40.
To find the range of the weight of Rosann's package, we need to determine which weight interval she falls into based on the given cost of $4.75.
1. We can start by deducting the base charge ($2.50) from the total cost ($4.75). This will give us the additional cost for the weight above 2 lb.
$4.75 - $2.50 = $2.25
2. Now, we need to find out how many $0.45 increments (the difference in costs between each weight interval) fit into the additional charge of $2.25. We divide $2.25 by $0.45 to get the number of increments.
$2.25 / $0.45 = 5
This means that Rosann's package falls into the weight range equivalent to 2 lb plus 5 increments of weight. Each increment corresponds to an increase of 0.5 lb.
So, the weight range of Rosann's package can be calculated as follows:
2 lb + (5 * 0.5 lb) = 2 lb + 2.5 lb = 4.5 lb
Therefore, the weight of Rosann's package falls within the range of 4.5 lb.
To determine the weight intervals and their corresponding costs, we need to identify the differences in cost between each interval.
Let's start with the first interval: 2 lb to less than 3 1/2 lb. The cost for this interval is $2.95.
Next, we move to the next interval: weights over 3 1/2 lb but less than 5 lb. The cost for this interval is $3.40.
By subtracting the cost of the first interval from the cost of the second interval, we can find the difference in cost between these two intervals:
$3.40 - $2.95 = $0.45
This means that for each additional interval beyond 3 1/2 lb, the cost increases by $0.45.
Now, let's work backwards to find the weight range for a cost of $4.75.
Since the cost for the interval from 3 1/2 lb to 5 lb is $3.40, we need to determine how many additional intervals we can include within the remaining cost of $4.75 - $3.40 = $1.35.
Dividing the remaining cost of $1.35 by the cost difference per interval of $0.45 will give us the number of additional intervals:
$1.35 / $0.45 = 3
Therefore, we can include 3 additional intervals.
Since each additional interval adds 1.5 lb to the weight, we multiply 3 by 1.5 to get the additional weight:
3 * 1.5 = 4.5 lb
Adding this to the starting weight of 3 1/2 lb, we can find the maximum weight:
3 1/2 lb + 4.5 lb = 8 lb
Therefore, the weight range for Rosann's package is from 3 1/2 lb to 8 lb.