Math

Question Shipping company IPS charges $0.75\$ 0.75 per pound for parcels 7\leq 7 lbs, $2.25\$ 2.25 per pound for 7<x197 < x \leq 19 lbs, and $3.25\$ 3.25 per pound for x19x \geq 19 lbs. There is a $3\$ 3 flat fee and $5\$ 5 fee for x19x \geq 19 lbs. Find the cost to ship a 33 lb package.

Studdy Solution

STEP 1

Assumptions
1. The shipping rates are piecewise functions of the weight of the parcel.
2. For packages weighing 7\leq 7 pounds, the charge is $0.75\$0.75 per pound.
3. For packages weighing >7> 7 pounds and 19\leq 19 pounds, the charge is $2.25\$2.25 per pound.
4. For packages weighing 19\geq 19 pounds, the charge is $3.25\$3.25 per pound.
5. There is a flat fee of $3\$3 for all deliveries.
6. There is an additional fee of $5\$5 for all deliveries weighing 19\geq 19 pounds.
7. IPS refuses to ship any parcel weighing more than 8282 pounds.
8. The weight of the package in question is 3333 pounds.

STEP 2

Define the piecewise function for the shipping costs C(x)C(x) based on the given intervals and rates.
C(x)={0.75x+3if 0<x72.25x+3if 7<x<193.25x+3+5if 19x82 C(x)=\left\{\begin{array}{ll} 0.75x + 3 & \text{if } 0 < x \leq 7 \\ 2.25x + 3 & \text{if } 7 < x < 19 \\ 3.25x + 3 + 5 & \text{if } 19 \leq x \leq 82 \end{array}\right.

STEP 3

Since the package weighs 3333 pounds, which falls into the third interval, we will use the formula for the interval 19x8219 \leq x \leq 82 to calculate the shipping cost.

STEP 4

Plug in the value of x=33x = 33 pounds into the formula for the third interval.
C(33)=3.2533+3+5C(33) = 3.25 \cdot 33 + 3 + 5

STEP 5

Calculate the cost by multiplying the weight of the package by the rate per pound.
C(33)=3.2533+8C(33) = 3.25 \cdot 33 + 8

STEP 6

Perform the multiplication.
C(33)=107.25+8C(33) = 107.25 + 8

STEP 7

Add the flat fee and the additional fee for heavy packages to get the total cost.
C(33)=107.25+8=115.25C(33) = 107.25 + 8 = 115.25
The cost to ship a package weighing 33 pounds with IPS is $115.25\$115.25.

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