Can you please help me solve this word problem?

Your boss has asked you to come up with a price model for one of the cell phone models that your company produces. In the model, costs and length of service are directly related. You are also asked to examine the profit formula based on selling a certain number of the phones to determine profitability based on how many are sold.

1. Suppose a cell phone service costs $45 per month, along with a $60 activation fee.
1. Write a cost equation that relates the number of months of service and the total cost for that time of service. (Hint: Let t = the number of months of service and c = the total cost.)
2. Use the cost equation to determine the number of months of service for a cost of $1,185.
2. Your company has determined that the profit equation (in thousands of dollars) of producing x thousand smartphones is as follows:

Profit = –x2 + 110x – 1,000

(If the number of smartphones is 40,000, for example, then x = 40). The break-even point is the number of smartphones sold and produced that would result in a profit of zero.
Use your preferred method for solving the quadratic equation to determine the break-even point.

Find the accumulated value of an investment of $16,000 at 4% compounded semiannually for 7 years.

$4,000 at 8% compunded annually for 50 years

Sure! Let's solve these word problems step by step:

1. Cost Equation:
To write a cost equation that relates the number of months of service (t) and the total cost (c), we need to consider the monthly cost and the activation fee.

Monthly Cost: $45
Activation Fee: $60

The total cost (c) can be calculated by multiplying the monthly cost by the number of months of service (t) and adding the activation fee:

c = 45t + 60

2. Determining the Number of Months of Service:
To find the number of months of service for a cost of $1,185, we can substitute the total cost (c) with 1,185 in the cost equation:

1,185 = 45t + 60

Now, let's solve for t:

1,185 - 60 = 45t
1,125 = 45t
t = 1,125 / 45
t = 25

Therefore, the number of months of service for a cost of $1,185 is 25 months.

3. Quadratic Equation and Break-Even Point:
The profit equation for producing x thousand smartphones is given as:

Profit = -x^2 + 110x - 1,000

The break-even point occurs when the profit is zero. In other words, we need to find the value of x that satisfies the equation "Profit = 0".

To determine the break-even point, we can set the profit equation equal to zero:

0 = -x^2 + 110x - 1,000

Now, we can solve this quadratic equation using a preferred method. One common method is factoring, but in this case, the equation is not easily factorable. So, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = -1, b = 110, and c = -1,000.

Substituting these values into the quadratic formula:

x = (-110 ± √(110^2 - 4(-1)(-1,000))) / (2(-1))

Simplifying further:

x = (-110 ± √(12,100 - 4,000)) / (-2)

x = (-110 ± √8,100) / (-2)

x = (-110 ± 90) / (-2) --> Taking both the positive and negative square roots

x1 = (-110 + 90) / (-2) = -20 / (-2) = 10
x2 = (-110 - 90) / (-2) = -200 / (-2) = 100

So, the break-even point occurs when x = 10 or x = 100. This means that if your company produces and sells either 10,000 or 100,000 smartphones, the profit will be zero.

I hope this explanation helps you solve the word problems. If you have any further questions, feel free to ask!