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Ludmilka [50]
3 years ago
13

Item 10 A bacterial culture quadruples in size every hour. You begin observing the number of bacteria 3 hours after the culture

is prepared. The amount y of bacteria x hours after the culture is prepared is represented by y=192(4x−3). When will there be 200,000 bacteria? Round your answer to the nearest hundredth. There will be 200,000 bacteria about hours after the culture is prepared.
Mathematics
1 answer:
Vinvika [58]3 years ago
5 0

Answer:

There will be 200 000 bacteria after 8 hours

Step-by-step explanation:

The crop quadruples every hour.

x represents the number of hours that pass after preparing the crop

y represents the number of bacteria after x hours.

The equation for y in function of x  3 hours after observing the crop is:

y = 192(4^{x - 3})

We want to know after how long there will be 200 000 bacteria.

Then we make y = 200 000 and clear x.

200\ 000 = 192(4^{x - 3})\\\\log(200\ 000) = log(192(4^{x-3}))\\\\log(200\ 000) = log(192) + log(4 ^{x-3})\\\\log(200\ 000) - log(192) = (x-3)log(4)\\\\\frac{log(200\ 000) - log(192)}{log(4)} = (x-3)\\\\x = 3 + \frac{log(200\ 000) - log(192)}{log(4)}\\\\x = 8\ hours

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Answer:

The smaller Number = 15

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Step-by-step explanation:

Let

The smaller number = x

The larger number = 5x - 8

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5x - 8 + x = 82

6x - 8 = 82

Add eight to both sides

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Therefore,

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= 75 - 8

= 67

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mezya [45]

Let's put more details in the figure to better understand the problem:

Let's first recall the three main trigonometric functions:

\text{ Sine }\theta\text{ = }\frac{\text{ Opposite Side}}{\text{ Hypotenuse}}\text{ Cosine }\theta\text{ = }\frac{\text{ Adjacent Side}}{\text{ Hypotenuse}}\text{ Tangent }\theta\text{ = }\frac{\text{ Opposite Side}}{\text{ Adjacent Side}}

For x, we will be using the Cosine Function:

\text{ Cosine }\theta\text{ = }\frac{\text{ Adjacent Side}}{\text{ Hypotenuse}}Cosine(45^{\circ})\text{ = }\frac{\text{ x}}{\text{ 1}6}(16)Cosine(45^{\circ})\text{ =  x}(16)(\frac{1}{\sqrt[]{2}})\text{ = x}\text{ }\frac{16}{\sqrt[]{2}}\text{ x }\frac{\sqrt[]{2}}{\sqrt[]{2}}\text{ = }\frac{16\sqrt[]{2}}{2}\text{ 8}\sqrt[]{2}\text{ = x}

Therefore, x = 8√2.

For y, we will be using the Sine Function.

\text{  Sine }\theta\text{ = }\frac{\text{ Opposite Side}}{\text{ Hypotenuse}}\text{ Sine }(45^{\circ})\text{ = }\frac{\text{ y}}{\text{ 1}6}\text{ (16)Sine }(45^{\circ})\text{ =  y}\text{ (16)(}\frac{1}{\sqrt[]{2}})\text{ = y}\text{ }\frac{16}{\sqrt[]{2}}\text{ x }\frac{\sqrt[]{2}}{\sqrt[]{2}}\text{ = }\frac{16\sqrt[]{2}}{2}\text{ 8}\sqrt[]{2}\text{ = y}

Therefore, y = 8√2.

5 0
2 years ago
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