Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points)
Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)
(10 points)
Answer:
Vertex = (12,64)
The meaning of this pair of values is that the y-coordinate is the maximum profit obtainable, and the x-coordinate is how many cups sold will make the maximum profit.
X-intercepts: 4 and 20
The meaning of theses values is that these amounts of cups sold (4 and 20) will make zero profit.
Step-by-step explanation:
To find the vertex we can use the formula for the x-coordinate of the vertex:
x_v = -b/2a
Where a and b are coefficients of the quadratic equation (in this case, a = -1 and b = 24)
So we have that:
x_v = -24 / (-2) = 12
The vertex is 12 cups of coffee. Now we apply this value to find the y-coordinate of the vertex:
f(x) = -12^2 + 24*12 - 80 = 64
So the vertex is (12,64). The meaning of this pair of values is that the y-coordinate is the maximum profit obtainable, and the x-coordinate is how many cups sold will make the maximum profit.
To find the x-intercepts, we need to make f(x) = 0 and find the values of x:
-x2 + 24x - 80 = 0
Delta = 24^2 - 80*4 = 256
sqrt(Delta) = 16
x1 = (-24 + 16)/(-2) = 4
x2 = (-24 - 16)/(-2) = 20
The x-intercepts are 4 and 20. The meaning of theses values is that these amounts of cups sold (4 and 20) will make zero profit.
arrive late. Valerie used the random-number table to find the experimental probability that of 5
flights, at least 2 will arrive late. The digit 0 represents flights arriving early. The digits 1,2,3,
4, 5, and 6 represent flights arriving on time. The digits 7, 8, and 9 represent flights arriving
late. Find the experimental probability that of 5 flights at least 2 will arrive late.
A: 3/10
B:2/5
C: 9/20
D: 11/20
Answer:
D. 11/20
Step-by-step explanation:
7, 8, and 9 represent late flights. We want to count how many of the 20 trials have at least two of these digits. Counting, we find that 11 of the 20 have at least two of these digits.
D. 11/20
miles.
Answer:
5 miles
Step-by-step explanation:
Think of this like a triangle. From the bottom of the tower, to the top of the tower, to the point 3 miles away, and back to the bottom of the tower.
So we already have 2 side lengths. The height of the tower, 3 miles, and the base, 4 miles. In order to find the 3rd length, the distance from the top of the tower to the point 4 miles away from the bottom, we need to apply the formula A squared + B squared = C squared.
We have A and B, (3 and 4) and we need C.
A squared (3 squared) is 9
B squared (4 squared) is 16
so 9 + 16 = C squared
9 + 16 = 25
C squared = 25
square root of 25 is 5
C = 5
The distance from the top of the tower to the point 4 miles away is 5.
By applying the Pythagorean theorem to the given problem, we find that the distance from the top of the tower to the point four miles away from the base of the tower is 5 miles.
Nimrod's problem is a classic application of the Pythagorean Theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the height of the tower is one side of the triangle (3 miles), and the distance from the base of the tower to the point Nimrod is interested in is the other side (4 miles). The distance from the top of the tower to that point is the hypotenuse.
Applying the Pythagorean theorem, we have: (Height of the tower)² + (Distance from the base to the point)² = (Distance from the top to the point)². So, this becomes: 3² + 4² = (Distance from the top to the point)². That simplifies into 9 + 16 = (Distance from the top to the point)², or 25 = (Distance from the top to the point)².
To find the actual distance (the length of the hypotenuse), we take the square root of 25, which is 5. Therefore, the distance from the top of the tower to the point four miles away from the base is 5 miles.
Using the Pythagorean theorem (a² + b² = c²), we can find the hypotenuse:
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25
c = 5
So the distance from the top of the tower to the point four miles away from the base is 5 miles.
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