PLEASE.
1. A frog jumps from a rock to the shore of a pond. Its path is given by the equation y=(-5/72)x^2+(5/3)x, where x is the horizontal distance in inches, and y is the height in inches. What is the frog's maximum height? How far had it traveled horizontally when it reached its maximum height?
2. A local bakery determines that when the price of a package of cookies is $12, it sells on average 220 packages of cookies per day. For every $0.50 increase in the price, it sells on average 10 packages of cookies less. Determine the price that maximizes the bakery's profit. Determine the maximum profit.
3.Find the quadratic function with the vertex (7,10) and with point (10,28) on its graph.
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1. To find the frog's maximum height, we need to find the vertex of the given quadratic equation. The vertex of a parabola in the form y = ax^2 + bx + c can be found using the formula x = -b/(2a). In this case, the equation is y = (-5/72)x^2 + (5/3)x, so a = -5/72 and b = 5/3.
To find the maximum height, we need to find the value of y when x is the horizontal distance at the vertex. Using the formula, x = -b/(2a), we get x = -(5/3) / (2 * (-5/72)) = 12.
Now, substitute the value of x back into the equation: y = (-5/72)(12^2) + (5/3)(12) = (-5/72)(144) + (60/3) = -10 + 60 = 50.
Therefore, the frog's maximum height is 50 inches.
To find the horizontal distance the frog had traveled when it reached its maximum height, we simply need to use the x-coordinate of the vertex, which is 12 inches.
Therefore, the frog had traveled 12 inches horizontally when it reached its maximum height.
2. To determine the price that maximizes the bakery's profit, we need to find the price at which the bakery sells the maximum number of packages of cookies. Since we know the relationship between the price and the number of packages sold, we can derive an equation to represent the number of packages sold as a function of the price.
Let P be the price of the package of cookies, and N be the number of packages sold per day. We are given that:
- When P is $12, N is 220 packages.
- For every $0.50 increase in P, N decreases by 10 packages.
Let's set up the equation:
N = -10(P - 12)/0.5 + 220
Simplifying the equation gives:
N = -20(P - 12) + 220
N = -20P + 460
Now, we can find the price that maximizes the bakery's profit by finding the vertex of this quadratic equation. The vertex of the parabola in the form y = ax^2 + bx + c is given by x = -b/(2a).
In this case, a = -20, b = 0 (since there is no coefficient for P), and c = 460.
Using the formula, x = -b/(2a), we get: x = -0 / (2 * (-20)) = 0.
Therefore, the price that maximizes the bakery's profit is $0.
To determine the maximum profit, we need to substitute the x-coordinate back into the quadratic equation. In this case, the equation is:
N = -20P + 460
Substituting x = 0, we get:
N = -20(0) + 460 = 460
Therefore, the maximum profit for the bakery is 460 packages of cookies.
3. To find the quadratic function, we can use the vertex form of a quadratic equation: y = a(x-h)^2 + k, where (h,k) represents the vertex of the parabola.
Given that the vertex is (7,10) and the point on the graph is (10,28), we can substitute these values into the equation:
28 = a(10-7)^2 + 10
Simplifying, we get:
28 = a(3)^2 + 10
28 = 9a + 10
9a = 18
a = 2
Therefore, the quadratic function with the given vertex (7,10) and point (10,28) is y = 2(x-7)^2 + 10.