Solved on Feb 05, 2024

Find the maximum number of graduated cylinders a scientist can purchase for $\$ 500$ , given the cost is $\$ 12.10$ each for 1 to 3, $\$ 11.07$ each for 4-47, and $\$ 10.20$ each for 48 or more.

STEP 1

Assumptions
1. The cost per graduated cylinder is $12.10 for quantities from 1 to 3.<br />2. The cost per graduated cylinder is$ 11.07 for quantities from 4 to 47.
3. The cost per graduated cylinder is $10.20 for quantities of 48 or more.<br />4. The total budget available for the purchase is$ 500.
5. The scientist wants to purchase the maximum number of cylinders within the budget.
6. The scientist can only purchase whole cylinders, not fractions of a cylinder.

STEP 2

First, we need to determine how many cylinders can be purchased at the highest price point, which is $12.10 per cylinder, without exceeding the budget.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor \frac{Total\, budget}{Cost\, per\, cylinder\, at\, \$12.10} \right\rfloor$

STEP 3

Calculate the maximum number of cylinders that can be purchased at $12.10 each with the given budget.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor \frac{\$500}{\$12.10} \right\rfloor$

STEP 4

Perform the division to find the number of cylinders.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor \frac{\$500}{\$12.10} \right\rfloor \approx \left\lfloor 41.32 \right\rfloor$

STEP 5

Since we cannot purchase a fraction of a cylinder, we take the floor of the result to find the whole number of cylinders.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor 41.32 \right\rfloor = 41$

STEP 6

Now, we need to check if purchasing 41 cylinders at $12.10 each is the most cost-effective option within the budget. We will compare this with purchasing 4 cylinders at$ 11.07 each and then additional cylinders at the same rate until the budget is exceeded.

STEP 7

Calculate the total cost for purchasing 4 cylinders at $11.07 each.
$Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07$

STEP 8

Perform the multiplication to find the total cost for 4 cylinders.
$Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07 = \$44.28$

STEP 9

Subtract the total cost for 4 cylinders from the budget to determine the remaining budget for additional cylinders at the rate of $11.07 each.
$Remaining\, budget = Total\, budget - Total\, cost\, for\, 4\, cylinders$

STEP 10

Calculate the remaining budget after purchasing 4 cylinders.
$Remaining\, budget = \$500 - \$44.28 = \$455.72$

STEP 11

Now, calculate the maximum number of additional cylinders that can be purchased at $11.07 each with the remaining budget.
$Number\, of\, additional\, cylinders\, at\, \$11.07 = \left\lfloor \frac{Remaining\, budget}{Cost\, per\, cylinder\, at\, \$11.07} \right\rfloor$

STEP 12

Perform the division to find the number of additional cylinders.
$Number\, of\, additional\, cylinders\, at\, \$11.07 = \left\lfloor \frac{\$455.72}{\$11.07} \right\rfloor \approx \left\lfloor 41.16 \right\rfloor$

STEP 13

Since we cannot purchase a fraction of a cylinder, we take the floor of the result to find the whole number of additional cylinders.
$Number\, of\, additional\, cylinders\, at\, \$11.07 = \left\lfloor 41.16 \right\rfloor = 41$

STEP 14

Add the initial 4 cylinders to the number of additional cylinders to find the total number of cylinders that can be purchased at the $11.07 rate.
$Total\, number\, of\, cylinders\, at\, \$11.07 = 4 + 41 = 45$

STEP 15

Now, we need to check if purchasing 45 cylinders at $11.07 each is the most cost-effective option within the budget. We will compare this with purchasing 48 cylinders at$ 10.20 each.

STEP 16

Calculate the total cost for purchasing 48 cylinders at $10.20 each.
$Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20$

STEP 17

Perform the multiplication to find the total cost for 48 cylinders.
$Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20 = \$489.60$

STEP 18

Since the total cost for 48 cylinders at $10.20 each is less than the budget, we need to determine if we can purchase additional cylinders at this rate with the remaining budget.

STEP 19

Subtract the total cost for 48 cylinders from the budget to determine the remaining budget for additional cylinders at the rate of $10.20 each.
$Remaining\, budget = Total\, budget - Total\, cost\, for\, 48\, cylinders$

STEP 20

Calculate the remaining budget after purchasing 48 cylinders.
$Remaining\, budget = \$500 - \$489.60 = \$10.40$

STEP 21

Since the remaining budget ( $10.40) is not enough to purchase an additional cylinder at$ 10.20 each, the maximum number of cylinders that can be purchased is 48.
The maximum number of graduated cylinders the scientist can purchase for $500 is 48.

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Find the maximum number of graduated cylinders a scientist can purchase for $500\$ 500$500, given the cost is $12.10\$ 12.10$12.10 each for 1 to 3, $11.07\$ 11.07$11.07 each for 4-47, and $10.20\$ 10.20$10.20 each for 48 or more.

STEP 1

STEP 2

STEP 3

Calculate the maximum number of cylinders that can be purchased at $12.10 each with the given budget.Number of cylinders at $12.10=⌊$500$12.10⌋Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor \frac{\$500}{\$12.10} \right\rfloorNumberofcylindersat$12.10=⌊$12.10$500​⌋

STEP 4

STEP 5

Since we cannot purchase a fraction of a cylinder, we take the floor of the result to find the whole number of cylinders.Number of cylinders at $12.10=⌊41.32⌋=41Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor 41.32 \right\rfloor = 41Numberofcylindersat$12.10=⌊41.32⌋=41

STEP 6

STEP 7

Calculate the total cost for purchasing 4 cylinders at $11.07 each.Total cost for 4 cylinders at $11.07=4×$11.07Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07Totalcostfor4cylindersat$11.07=4×$11.07

STEP 8

Perform the multiplication to find the total cost for 4 cylinders.Total cost for 4 cylinders at $11.07=4×$11.07=$44.28Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07 = \$44.28Totalcostfor4cylindersat$11.07=4×$11.07=$44.28

STEP 9

STEP 10

Calculate the remaining budget after purchasing 4 cylinders.Remaining budget=$500−$44.28=$455.72Remaining\, budget = \$500 - \$44.28 = \$455.72Remainingbudget=$500−$44.28=$455.72

STEP 11

STEP 12

STEP 13

STEP 14

Add the initial 4 cylinders to the number of additional cylinders to find the total number of cylinders that can be purchased at the $11.07 rate.Total number of cylinders at $11.07=4+41=45Total\, number\, of\, cylinders\, at\, \$11.07 = 4 + 41 = 45Totalnumberofcylindersat$11.07=4+41=45

STEP 15

STEP 16

Calculate the total cost for purchasing 48 cylinders at $10.20 each.Total cost for 48 cylinders at $10.20=48×$10.20Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20Totalcostfor48cylindersat$10.20=48×$10.20

STEP 17

Perform the multiplication to find the total cost for 48 cylinders.Total cost for 48 cylinders at $10.20=48×$10.20=$489.60Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20 = \$489.60Totalcostfor48cylindersat$10.20=48×$10.20=$489.60

STEP 18

Since the total cost for 48 cylinders at $10.20 each is less than the budget, we need to determine if we can purchase additional cylinders at this rate with the remaining budget.

STEP 19

STEP 20

Calculate the remaining budget after purchasing 48 cylinders.Remaining budget=$500−$489.60=$10.40Remaining\, budget = \$500 - \$489.60 = \$10.40Remainingbudget=$500−$489.60=$10.40

STEP 21

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Find the maximum number of graduated cylinders a scientist can purchase for $\$ 500$ , given the cost is $\$ 12.10$ each for 1 to 3, $\$ 11.07$ each for 4-47, and $\$ 10.20$ each for 48 or more.

Calculate the maximum number of cylinders that can be purchased at $12.10 each with the given budget.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor \frac{\$500}{\$12.10} \right\rfloor$

Since we cannot purchase a fraction of a cylinder, we take the floor of the result to find the whole number of cylinders.
$Number\, of\, cylinders\, at\, \$12.10 = \left\lfloor 41.32 \right\rfloor = 41$

Calculate the total cost for purchasing 4 cylinders at $11.07 each.
$Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07$

Perform the multiplication to find the total cost for 4 cylinders.
$Total\, cost\, for\, 4\, cylinders\, at\, \$11.07 = 4 \times \$11.07 = \$44.28$

Calculate the remaining budget after purchasing 4 cylinders.
$Remaining\, budget = \$500 - \$44.28 = \$455.72$

Add the initial 4 cylinders to the number of additional cylinders to find the total number of cylinders that can be purchased at the $11.07 rate.
$Total\, number\, of\, cylinders\, at\, \$11.07 = 4 + 41 = 45$

Calculate the total cost for purchasing 48 cylinders at $10.20 each.
$Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20$

Perform the multiplication to find the total cost for 48 cylinders.
$Total\, cost\, for\, 48\, cylinders\, at\, \$10.20 = 48 \times \$10.20 = \$489.60$

Calculate the remaining budget after purchasing 48 cylinders.
$Remaining\, budget = \$500 - \$489.60 = \$10.40$