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lianna [129]
3 years ago
11

The parabola y=ax^2+bx-10 passes through the points (4.5,8) and (-2.5, 15). Determine the values of a and b

Mathematics
1 answer:
yuradex [85]3 years ago
6 0
y=ax^2+bx-10 \\ \\
(4.5,8) \\
8=a \times 4.5^2 + b \times 4.5-10 \\
8=20.25a+4.5b-10 \ \ \ |+10 \\
18=20.25a+4.5b \\ \\
(-2.5,15) \\
15=a \times (-2.5)^2+b \times (-2.5)-10 \\
15=6.25a-2.5b-10 \ \ \ |+10 \\
25=6.25a-2.5b

The system of equations:
18=20.25a+4.5b \ \ \ |\times 5 \\
25=6.25a-2.5b \ \ \ |\times 9 \\ \\
90=101.25a+22.5b \\
\underline{225=56.25a-22.5b} \\
90+225=101.25a+56.25a \\
315=157.5a \ \ \ |\div 157.5 \\
a=2 \\ \\
25=6.25a-2.5b \\
25=6.25 \times 2-2.5b \\
25=12.5 -2.5b \ \ \ |-12.5 \\
12.5=-2.5b \ \ \ |\div (-2.5) \\
b=-5 \\ \\
\boxed{a=2} \\ \boxed{b=-5} \\ \boxed{y=2x^2-5x-10}
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602 tourists visited only the LEGOLAND.

Step-by-step explanation:

To solve this problem, we must build the Venn's Diagram of this set.

I am going to say that:

-The set A represents the tourists that visited LEGOLAND

-The set B represents the tourists that visited Universal Studios

-The set C represents the tourists that visited Magic Kingdown.

-The value d is the number of tourists that did not visit any of these parks, so: d = 58

We have that:

A = a + (A \cap B) + (A \cap C) + (A \cap B \cap C)

In which a is the number of tourists that only visited LEGOLAND, A \cap B is the number of tourists that visited both LEGOLAND and Universal Studies, A \cap C is the number of tourists that visited both LEGOLAND and the Magic Kingdom. and A \cap B \cap C is the number of students that visited all these parks.

By the same logic, we have:

B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)

C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)

This diagram has the following subsets:

a,b,c,d,(A \cap B), (A \cap C), (B \cap C), (A \cap B \cap C)

There were 1,107 tourists suveyed. This means that:

a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107

We start finding the values from the intersection of three sets.

The problem states that:

36 tourists had visited all three theme parks. So:

(A \cap B \cap C) = 36

72 tourists had visited both LEGOLAND and Universal Studios. So:

(A \cap B) + (A \cap B \cap C) = 72

(A \cap B) = 72 - 36

(A \cap B) = 36

79 tourists had visited both the Magic Kingdom and Universal Studios

(B \cap C) + (A \cap B \cap C) = 79

(B \cap C) = 79 - 36

(B \cap C) = 43

68 tourists had visited both the Magic Kingdom and LEGOLAND

(A \cap C) + (A \cap B \cap C) = 68

(A \cap C) = 68 - 36

(A \cap C) = 32

258 tourists had visited Universal Studios:

B = 258

B = b + (B \cap C) + (A \cap B) + (A \cap B \cap C)

258 = b + 43 + 36 + 36

b = 143

268 tourists had visited the Magic Kingdom:

C = 268

C = c + (A \cap C) + (B \cap C) + (A \cap B \cap C)

268 = c + 32 + 43 + 36

c = 157

How many tourists only visited the LEGOLAND (of these three)?

We have to find the value of a, and we can do this by the following equation:

a + b + c + d + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 1,107

a + 143 + 157 + 58 + 36 + 32 + 43 + 36 = 1,107

a = 602

602 tourists visited only the LEGOLAND.

6 0
3 years ago
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