how would i do this quadratic word problem
A local daycare centre charges $65 per day to care for a child. The daycare currently cares for 150 children per day. A survey shows that the enrollment in the daycare will increase by 10 children for each $5 decrease in daily fee, and would decrease similarly if the daily fees are increased.
What daily fee would result in the greatest revenue for the daycare centre?
What is your school SUBJECT?
sorry i forgot to write math
To solve this quadratic word problem, we need to determine the daily fee that would result in the greatest revenue for the daycare center.
Let's break down the problem into steps:
Step 1: Determine the relationship between the number of children and the daily fee. The problem states that for each $5 decrease in the daily fee, the enrollment in the daycare will increase by 10 children. Similarly, if the daily fees are increased, the enrollment will decrease similarly.
Step 2: Create a mathematical expression based on the given information. Let's use the variables "x" to represent the daily fee and "n" to represent the number of children. Based on the given information, we can express the relationship between the daily fee and the number of children as:
n = 150 + 10((65 - x)/5)
Here, (65 - x)/5 represents the number of $5 decreases from the initial fee of $65. Multiplying it by 10 gives the corresponding number of children for each $5 decrease.
Step 3: Calculate the revenue. Revenue is the product of the daily fee (x) and the number of children (n). Hence, the revenue function (R) can be expressed as:
R = xn
Step 4: Simplify the revenue function. Substituting the value of n from Step 2 into the revenue function, we get:
R = x(150 + 10((65 - x)/5))
Simplifying further:
R = x(150 + 2(65 - x))
R = x(280 - 2x)
R = 280x - 2x^2
Step 5: Determine the maximum/minimum point. We can find the value of x at which the revenue is maximized by finding the vertex of the quadratic equation R = 280x - 2x^2. The x-coordinate of the vertex can be calculated using the formula x = -b/(2a), where a = -2 and b = 280 in this equation.
x = -280/(2(-2))
x = -280/(-4)
x = 70
Step 6: Interpret the answer. The daily fee that would result in the greatest revenue for the daycare center is $70.
Therefore, setting the daily fee to $70 would maximize the revenue for the daycare center.