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3 years ago

Find a positive number such that the sum of the number and its reciprocal is as small as possible?

1 answer:

5 0

You're minimizing the function $f(x)=x+\dfrac1x$ where $x>0$ .

Take the derivative, set to zero and find the critical points:

$f'(x)=1-\dfrac1{x^2}=0\implies1=\dfrac1{x^2}$

This has two roots, $x=\pm1$ , but we ignore the negative root.

Check the signs of the derivative to the left and right of $x=1$ and you'll see that $f'(x)$ for $x$ , and $f'(x)>0$ for $x>1$ , which means a minimum occurs at $x=1$ .

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What is the solution to the inequality below?

Svetlanka [38]

Answer:

B. x < -8 or x > 8

Step-by-step explanation:

You can use process of elimination to solve this problem by going through every solution and testing them out, but let's jump right to B.

Process:

You know that since the inequality states that x^2 has to be greater than 64, x has to be more than 8, or less than -8.

This is because 8^2 = 64, and -8^2 = 64, and the inequality requires the answer to be more than 64.

Looking at B., you can see that if x is < -8, the square of, for example, -9, would be 81. This is greater than 64, so this works!

Now, B. also has an alternative. The 'or' is a major clue to which is the correct answer, since the square root of any number can be positive or negative. (-8^2 = 8^2)

The 'or' states that x must be greater than 8. So, for example, if we take the square of 10, we get 100, and that is also greater than 64.

We've proven that this solution is accurate for both parts, so it is definitely the one we want!

Hope this helps!

4 0

3 years ago

Round 8.36 to the nearest one decimal place

BartSMP [9]

8.4 because 6 would round up to 10 and 10+30 =40 for 8.40 or just 8.4

4 0

3 years ago

3/7 of 8th graders play sports. is 3/5 of the 8th graders that play sports are boys, what fraction of the 8th graders are boys

Svetach [21]

Answer:

$\dfrac{9}{35}$ is the fraction of 8th graders who are boys.

Step-by-step explanation:

Let total number of 8th graders = $x$

Given that, $\frac{3}{7}$ of 8th graders play sports

$\Rightarrow$ 8th graders who play sports = $\frac{3}{7} \times x$

Also, Given that, $\frac{3}{5}$ of 8th graders that play sports are boys

Number of 8th graders that play sports, and are boys = $\frac{3}{5} \times \text{Number of 8th graders who play sports}$

$\Rightarrow \dfrac{3}{5} \times \dfrac{3}{7} \times x\\\Rightarrow \dfrac{9}{35} \times x$

Hence, the answer is:

$\dfrac{9}{35}$ is the fraction of 8th graders who are boys.

6 0

3 years ago

An 8-sided fair die is rolled twice and the product of the two numbers obtained when the die is rolled two times is calculated.

SCORPION-xisa [38]

Answer:

(a)See below

(b)I) 0.125 (ii)0.828125 (iii)0.234375 (iv) 0.625 (v)0.65625

Step-by-step explanation:

When an 8-sided die is rolled twice, the sample space is the set of all possible pairs (a,b) where a is the 1st outcome and b is the 2nd outcome.

The sample space is:

$[(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),(1, 7),(1, 8)\\(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),(2, 7),(2, 8)\\(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),(3, 7),(3, 8)\\(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),(4, 7),(4, 8)\\(5, 1), (5, 2), (5, 3), (5, 4), (5, 5),(5, 6),(5, 7),(5, 8)\\(6, 1), (6, 2), (6, 3), (6, 4)(6, 5),(6, 6),(6, 7),(6, 8)\\(7, 1), (7, 2), (7, 3), (7, 4)(7, 5),(7, 6),(7, 7),(7, 8)\\(8, 1), (8, 2), (8, 3), (8, 4)(8, 5),(8, 6),(8, 7),(8, 8)]$

PART A

The sample space of the product a*b of each pair forms the required possibility diagram.

This is given as:

$1, 2, 3, 4, 5, 6,7,8\\2, 4, 6, 8, 10, 12,14,16\\3,6,9,12,15,18,21,24\\4,8,12,16,20,24,28,32\\5,10,15,20,25,30,35,40\\6,12,18,24,30,36,42,48\\7,14,21,28,35,42,49,56\\8,16,24,32,40,48,56,64$

PART B

I) A prime number

The number of products that gives prime numbers = 8

The probability that the product is a prime number

$=\dfrac{8}{64}= \dfrac{1}{8}\\=0.125$

ii) Not a perfect square

Number of products that results in perfect squares =11

The probability that the product is not a perfect square

$=\dfrac{64-11}{64}= \dfrac{53}{64}\\=0.828125$

iii) A multiple of 5

Number of products that are multiples of 5=15

The probability that the product is a multiple of 5

$=\dfrac{15}{64}\\=0.234375$

iv) Less than or equal to 21

Number of products which are less than or equal to 21=40

The probability that the product less than or equal to 21

$=\dfrac{40}{64}=\dfrac{5}{8}\\=0.625$

v) Divisible by 4 or 6

Number of products divisible by 4 or 6= 42

The probability that the product is divisible by 4 or 6

$=\dfrac{42}{64}=\dfrac{21}{32}\\=0.65625$

8 0

3 years ago

a section of ground measures 1.5 cubic meters and contains 0.78 gram of gold. what is the amount of gold per cubic meter? explai

I am Lyosha [343]

The answer

the amount of gold per cubic meter can be found by the ratio

0.78 g /1 .5 cubic meters = 0.52 g / cubic meter,
proof
it was said the amount of gold per cubic meter (Density)

7 0

3 years ago