An electronics company has developed a new hand-held device. The company predicts that the start-up cost to manufacture the new product will be $125,000, and the cost to make one device will be $6.50.
A. If the company plans on selling the devices at a price of $9, write and solve an inequality to determine how many must be sold for the company to make a profit.
B. The cost of one device is 10% more than the company predicted. What is the new cost of making one device? How many devices must they now sell at the same price to make a profit?
C. Suppose the company wants to start making a profit after selling the same number of devices you found in part a. What is the lowest the new price can be? Explain how you found this price. HELP!!!!
A. To determine how many devices must be sold for the company to make a profit, we can set up an inequality. Let's represent the number of devices sold as "x".
The revenue from selling x devices at a price of $9 per device can be expressed as:
Revenue = 9x
The cost of manufacturing x devices is given by:
Cost = 6.50x + 125,000
To make a profit, the revenue should be greater than the cost. Therefore, we can write the inequality as:
9x > 6.50x + 125,000
Now, let's solve the inequality to find the minimum number of devices that need to be sold for the company to make a profit:
9x - 6.50x > 125,000
2.5x > 125,000
x > 125,000 / 2.5
x > 50,000
So, the company must sell more than 50,000 devices to make a profit.
B. If the cost of making one device is 10% more than the predicted cost of $6.50, we can calculate the new cost as follows:
New cost = $6.50 + ($6.50 * 10%) = $6.50 + ($6.50 * 0.10) = $6.50 + $0.65 = $7.15
Now, we can use the same inequality as before to determine how many devices must be sold to make a profit. However, the cost value will be updated. So, the inequality becomes:
9x > 7.15x + 125,000
Solving this inequality will give us the new minimum number of devices that need to be sold for the company to make a profit.
C. To find the lowest the new price can be while still starting to make a profit after selling the same number of devices found in part A, we need to use a similar approach as before.
Let's represent the new price as "p". The revenue will now be:
Revenue = px
The cost will remain the same as before:
Cost = 6.50x + 125,000
To start making a profit, the revenue should be greater than the cost. Therefore, we can write the new inequality as:
px > 6.50x + 125,000
Simplifying this inequality will help us find the lowest possible new price.
I hope this explanation helps you! Let me know if you have any other questions.
A. To determine the number of devices that must be sold for the company to make a profit, we need to consider the start-up cost and the cost to make one device, along with the selling price.
Let's denote the number of devices sold as "x". So, the cost to manufacture "x" devices is given by:
Cost to manufacture "x" devices = Cost to make one device × Number of devices = $6.50x
The revenue from selling "x" devices is given by:
Revenue from selling "x" devices = Selling price × Number of devices = $9x
To determine the profit, we subtract the cost to manufacture from the revenue:
Profit = Revenue - Cost to manufacture = $9x - $6.50x = $2.50x
The company will make a profit if the profit is positive. So, we need to solve the inequality:
$2.50x > $125,000
Dividing both sides of the inequality by $2.50, we get:
x > $125,000 / $2.50
x > 50,000
Therefore, the company must sell more than 50,000 devices to make a profit.
B. If the cost of one device is 10% more than the predicted cost of $6.50, then the new cost of making one device can be calculated as follows:
New cost of making one device = $6.50 + ($6.50 × 10%) = $6.50 + $0.65 = $7.15
Using the same revenue and profit calculation as before, the profit equation becomes:
Profit = $9x - $7.15x = $1.85x
To find the new number of devices that must be sold to make a profit, we need to solve the inequality:
$1.85x > $125,000
Dividing both sides of the inequality by $1.85, we get:
x > $125,000 / $1.85
x > 67,567.57 (approx.)
Therefore, the company must sell more than 67,567 devices (approx.) at the same price of $9 to make a profit.
C. To determine the lowest new price at which the company wants to start making a profit after selling the same number of devices found in part A, we can divide the total cost by the number of devices and add the desired profit per device.
Original cost per device = $6.50
Number of devices = 50,000 (as found in part A)
Total cost = Start-up cost + (Cost to make one device × Number of devices) = $125,000 + ($6.50 × 50,000) = $400,000
Let's denote the new price as "p". To make a profit, we need to have:
Profit per device = Selling price - Cost per device ≥ $0 (positive profit)
Setting up the equation:
p - ($6.50 + p) ≥ 0
p - $6.50 - p ≥ 0
-$6.50 ≥ -p
$p ≥ $6.50
Therefore, the lowest new price that the company can set to start making a profit after selling the same number of devices found in part A is $6.50.
By maintaining the same selling price as before ($9), the company will already make a profit, as shown in part A.
The cost to make x items will be
c(x) = 125000+6.50x
So, for
(A) you want 9x >= 125000+6.50x
see what yo can do with the other parts.