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3 years ago

Complete the square to determine the minimum or maximum value of the function defined by the expression.

Mathematics

1 answer:

Alekssandra [29.7K]3 years ago

3 0

The minimum value at -1 ⇒ D

Step-by-step explanation:

The completing square form of ax² + bx + c is a(x - h)² + k, where

$h=\frac{-b}{2a}$
k is the value of of the expression when x = h
k is minimum if a > 0 and maximum if a < 0

∵ The expression is x² + 4x + 3

∴ a = 1 , b = 4 , c = 3

- Use the rule above to find h

∵ $h=\frac{-4}{2(1)}$

∴ h = -2

- To find k substitute x by the value of h

∵ k = (-2)² + 4(-2) + 3 = 4 - 8 + 3

∴ k = -1

- Substitute h and k in the form of the completing square

∵ a(x - h)² + k = 1(x - -2)² + (-1)

∴ a(x - h)² + k = (x + 2)² - 1

∴ x² + 4x + 3 = (x + 2)² - 1

∵ The completing square is (x + 2)² - 1

∵ a = 1 ⇒ greater than zero

∴ The value is minimum

- The minimum value is the value of k

∵ k = -1

∴ The minimum value of the function is -1

The minimum value at -1

Learn more:

You can learn more about the quadratic function in brainly.com/question/9390381

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Answer:

a) 0.7291 is the probability that more than half out of 10 vehicles carry just 1 person.

b) 0.996 is the probability that more than half of the vehicles carry just one person.

Step-by-step explanation:

We are given the following information:

A) Binomial distribution

We treat vehicle on road with one passenger as a success.

P(success) = 64% = 0.64

Then the number of vehicles follows a binomial distribution, where

$P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}$

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 10

We have to evaluate:

$P(x \geq 6) = P(x =6) +...+ P(x = 10) \\= \binom{10}{6}(0.64)^6(1-0.64)^4 +...+ \binom{10}{10}(0.64)^{10}(1-0.79)^0\\=0.7291$

0.7291 is the probability that more than half out of 10 vehicles carry just 1 person.

B) By normal approximation

Sample size, n = 92

p = 0.64

$\mu = np = 92(0.64) = 58.88$

$\sigma = \sqrt{np(1-p)} = \sqrt{92(0.64)(1-0.64)} = 4.60$

We have to evaluate the probability that more than 47 cars carry just one person.

$P(x \geq 47)$

After continuity correction, we will evaluate

$P( x \geq 46.5) = P( z > \displaystyle\frac{46.5 - 58.88}{4.60}) = P(z > -2.6913)$

$= 1 - P(z \leq -2.6913)$

Calculation the value from standard normal z table, we have,

$P(x > 46.5) = 1 - 0.004 = 0.996 = 99.6\%$

0.996 is the probability that more than half out of 92 vehicles carry just one person.

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