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krek1111 [17]
3 years ago
9

Please help with 15, 17 and 19

Mathematics
1 answer:
Irina-Kira [14]3 years ago
6 0

Given:

15. \log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)

17. \log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)

19. 2^{\log_2100}

To find:

The values of the given logarithms by using the properties of logarithms.

Solution:

15. We have,

\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)

Using property of logarithms, we get

\log_{\frac{1}{2}}\left(\dfrac{1}{2}\right)=1         [\because \log_aa=1]

Therefore, the value of \log_{\frac{1}{2}}\left(\dfrac{1}{2}\right) is 1.

17. We have,

\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)

Using properties of logarithms, we get

\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)=-\log_{\frac{3}{4}}\left(\dfrac{3}{4}\right)                    [\because \log_a\dfrac{m}{n}=-\log_a\dfrac{n}{m}]

\log_{\frac{3}{4}}\left(\dfrac{4}{3}\right)=-1                 [\because \log_aa=1]

Therefore, the value of \log_{\frac{3}{4}}\left(\dfrac{4}{3}\right) is -1.

19. We have,

2^{\log_2100}

Using property of logarithms, we get

2^{\log_2100}=100          [\because a^{\log_ax}=x]

Therefore, the value of 2^{\log_2100} is 100.

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Find the 8th term of the geometric sequence whose common ratio is 2/3 and whose first term is 7​
motikmotik

Answer:

The 8th term of the sequence is 896/2187.

Step-by-step explanation:

We want to find the 8th term of a geometric sequence whose common ratio is 2/3 and whose first term is 7.

We can write a direct formula. Recall that the direct formula of a geometric sequence is given by:

\displaystyle x_{n} = a\left(r\right)^{n-1}

Where <em>a</em> is the initial term and <em>r</em> is the common ratio.

Substitute:

\displaystyle x_{n} = 7\left(\frac{2}{3}\right)^{n-1}

To find the 8th term, let <em>n</em> = 8. Substitute and evaluate:

\displaystyle \begin{aligned} x_{8} &= 7\left(\frac{2}{3}\right)^{(8) - 1} \\ \\ &= 7\left(\frac{2}{3}\right)^{7} \\ \\ &= 7\left(\frac{128}{2187}\right) \\ \\ &= \frac{896}{2187} = 0.4096...\end{aligned}

In conclusion, the 8th term of the sequence is 896/2187.

8 0
2 years ago
One die is blue and the other die is yellow. Both are rolled. What is the probability of getting the shm of 8?
suter [353]

Answer:

5 /36

Step-by-step explanation:

Sample space :

Number of faces^number of dice = 6^2 = 36

Sum of 8 on a roll of 2 dies :

Probability = required outcome / Total possible outcomes

Required outcome = number of times a sum of 8 is obtained on a roll of 2 dies = 5

Total possible outcomes = sample space = 36

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A printer has a contract to print 100,000 posters for a political candidate. He can run the posters by using any number of plate
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Answer:

25

Step-by-step explanation:

From the given information;

Numbers of posters that can be printed in an hour = no of impression/hour × no of plate utilized in each impression.

= 1000x

Thus, the required number of hours it will take can be computed as:

\implies \dfrac{100000}{1000x} \\ \\ =\dfrac{100}{x}

cost per hour = 125

If each plate costs $20 to make, then the total number of plate will equal to 40x

∴

The total cost can be computed as:

C(x) = (\dfrac{100}{x}) \times 125 + 20 x --- (1)

C'(x) = (-\dfrac{12500}{x^2})  + 20  --- (2)

At C'(x) = 0

\dfrac{12500}{x^2} = 20

\dfrac{12500}{20} = x^2

x^2= 625

x = \sqrt{625}

x = 25

C'' (x) = -12500 \times \dfrac{-2}{x^3} +0

C'' (x) = \dfrac{25000}{x^3}

where; x = 25

C'' (x) = \dfrac{25000}{25^3}

C''(x) = 1.6

Thus, at x = 25, C'' > 0

As such, to minimize the cost, the printer needs to make 25 metal plates.

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