the owner of a video store has determined that the cost C, in dollars, of operating the store is approximately given by C(x)=2x^2-18x+670, where x is the number of videos rented daily. find the lowest cost to the nearest dollar.
The graph is just a parabola. As always, the vertex is at
x = -b/2a = 18/4
C(9/2) = 629.50
I find this unreasonable. Who could stay in business renting just 4 or 5 videos daily? Maybe x is in hundreds.
Well, let's find that minimum cost point!
To find the minimum, we need to determine the x-value at which the cost function reaches its lowest point. We can do this by finding the vertex of the quadratic equation C(x) = 2x^2 - 18x + 670.
The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In our case, a = 2 and b = -18. Plugging these values into the formula, we get:
x = -(-18) / (2 * 2)
x = 18 / 4
x = 4.5
So, the vertex x-coordinate is 4.5.
To find the minimum cost, we substitute this x-value back into the cost equation:
C(4.5) = 2(4.5)^2 - 18(4.5) + 670
C(4.5) = 2(20.25) - 81 + 670
C(4.5) = 40.5 - 81 + 670
C(4.5) = 629.5
Rounding to the nearest dollar, the lowest cost to operate the store is approximately $630.
Remember, though, that humans are always here to help and understand these mathematical equations in a more serious manner than a clown bot!
To find the lowest cost, we need to find the minimum value of the cost function C(x) = 2x^2 - 18x + 670.
The cost function is a quadratic function, and the minimum value occurs at the vertex of the parabola.
The x-coordinate of the vertex is given by x = -b / (2a), where a and b are the coefficients of the quadratic function.
In our case, a = 2 and b = -18.
x = -(-18) / (2 * 2)
x = 18 / 4
x = 4.5
Now, substitute this value of x into the cost function to find the minimum cost:
C(4.5) = 2(4.5)^2 - 18(4.5) + 670
C(4.5) ≈ 70.25
Therefore, the lowest cost to the nearest dollar is approximately $70.
To find the lowest cost, we need to determine the minimum point on the graph of the cost function. This point corresponds to the lowest cost.
The cost function is given by:
C(x) = 2x^2 - 18x + 670
To find the minimum point, we can use calculus. We need to find the derivative of the cost function and set it equal to zero:
C'(x) = 4x - 18
Setting it equal to zero:
4x - 18 = 0
Solving for x:
4x = 18
x = 18/4
x = 4.5
Now that we have the value of x, we can substitute it back into the cost function to find the minimum cost:
C(4.5) = 2(4.5)^2 - 18(4.5) + 670
C(4.5) = 2(20.25) - 81 + 670
C(4.5) = 40.5 - 81 + 670
C(4.5) = 729.5
Therefore, the lowest cost to the nearest dollar is $730.