Consider the supply equation where x is the quantity supplied in units of a thousand and p is the unit price in dollars.
p=x^3+x+10
Determine the price at which the supplier will make 2,500 units of the commodity available in the market. (Give your answer correct to the nearest cent.)
$ ??
P = x^3 + x + 10
P = 2.5^3 + 2.5 + 10 = $28.13
Well, let's substitute x = 2.5 into the equation and solve for p:
p = (2.5)^3 + 2.5 + 10
p ≈ 2.375 + 2.5 + 10
p ≈ 14.875
So, the price at which the supplier will make 2,500 units of the commodity available in the market is approximately $14.88. Keep in mind that due to rounding, there might be a small difference in the actual value.
To determine the price at which the supplier will make 2,500 units of the commodity available in the market, we need to substitute x = 2.5 into the supply equation, p = x^3 + x + 10.
p = (2.5)^3 + 2.5 + 10
p = 6.25 + 2.5 + 10
p = 18.75 + 10
p = 28.75
Therefore, the price at which the supplier will make 2,500 units of the commodity available in the market is $28.75.
To determine the price at which the supplier will make 2,500 units available in the market, we need to solve the equation p = 2,500.
Substitute the value of p into the supply equation:
2500 = x^3 + x + 10
Now, we need to solve this equation for x. However, finding the exact solution to this cubic equation can be complex and time-consuming.
One way to solve this is by using numerical methods or a graphing calculator. Let's use the "guess and check" method with a graphing calculator.
If you have access to a graphing calculator, follow these steps:
1. Graph the equation y = 2500.
2. Graph the equation y = x^3 + x + 10.
3. Observe the point(s) of intersection between the two graphs.
4. Find the x-coordinate of the intersection point(s), which represents the value of x that satisfies the equation.
Alternatively, we can use a spreadsheet software like Microsoft Excel or Google Sheets to set up a table and find approximate values for x.
Set up a table with two columns. In the first column, list some values for x (e.g., -10, -5, 0, 5, 10). In the second column, calculate the corresponding values of p using the supply equation p = x^3 + x + 10. Keep adjusting the values of x until you get a corresponding value of p that is close to 2,500.
Once you find the value of x that gives a value of p close to 2,500, substitute that value into the supply equation p = x^3 + x + 10 to calculate the exact price.
Keep in mind that these methods provide an approximate answer, so the exact price may not be achieved.