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finlep [7]
3 years ago
9

Suppose that the height (in centimeters) of a candle is a linear function of the amount of time (in hours) it has been burning.

After 6 hours of burning, a candle has a height of 17.4 centimeters. After 23 hours of burning, its height is 7.2 centimeters. What is the height of the candle after 11 hours?
Mathematics
1 answer:
Alborosie3 years ago
7 0
Assume that the rule connecting height of the candle to time is a linear one.  If you do, then we have to find the equation of this line, and then use this equation to predict the height of the candle after 11 hours.

Two points on this line are (6,17.4) and (23, 7.2).  The slope is thus
         7.2-17.4           -6
m = ---------------  =  -----------  or   -3/5.
           23-6                 10

Find the equation of the line.  I'm going to use the slope-intercept formula:

y = mx + b  =>   7.2 = (-3/5)(23) + b.  Solving for b,   b = 21.

Now    we know that y = (-3/5)x + 21

Let x=11 to predict the height of the candle at that time.

y = (-3/5)(11) + 21 = 14.4 inches  (answer)
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Klio2033 [76]

First part of question:

Find the general term that represents the situation in terms of k.

The general term for geometric series is:

a_{n}=a_{1}r^{n-1}

a_{1} = the first term of the series

r = the geometric ratio

a_{1} would represent the height at which the ball is first dropped. Therefore:

a_{1} = k

We also know that the ball has a rebound ratio of 75%, meaning that the ball only bounces 75% of its original height every time it bounces. This appears to be our geometric ratio. Therefore:

r=\frac{3}{4}

Our general term would be:

a_{n}=a_{1}r^{n-1}

a_{n}=k(\frac{3}{4}) ^{n-1}

Second part of question:

If the ball dropped from a height of 235ft, determine the highest height achieved by the ball after six bounces.

k represents the initial height:

k = 235\ ft

n represents the number of times the ball bounces:

n = 6

Plugging this back into our general term of the geometric series:

a_{n}=k(\frac{3}{4}) ^{n-1}

a_{n}=235(\frac{3}{4}) ^{6-1}

a_{n}=235(\frac{3}{4}) ^{5}

a_{n}=55.8\ ft

a_{n} represents the highest height of the ball after 6 bounces.

Third part of question:

If the ball dropped from a height of 235ft, find the total distance traveled by the ball when it strikes the ground for the 12th time. ​

This would be easier to solve if we have a general term for the <em>sum </em>of a geometric series, which is:

S_{n}=\frac{a_{1}(1-r^{n})}{1-r}

We already know these variables:

a_{1}= k = 235\ ft

r=\frac{3}{4}

n = 12

Therefore:

S_{n}=\frac{(235)(1-\frac{3}{4} ^{12})}{1-\frac{3}{4} }

S_{n}=\frac{(235)(1-\frac{3}{4} ^{12})}{\frac{1}{4} }

S_{n}=(4)(235)(1-\frac{3}{4} ^{12})

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8 0
3 years ago
How can you write the expression with rationalized denominator? sqrt3 - sqrt6 / sqrt 3 + sqrt 6
harina [27]
Answer:  2√2 - 3

Explanation:


The expession written properly is:

\frac{ \sqrt{3}- \sqrt{6}  }{ \sqrt{3}+ \sqrt{6} }

To rationalize that kind of expressions, this is to eliminate the radicals on the denominator you use conjugate rationalization.

That is, you have to multiply both numerator and denominator times the conjugate of the denominator.

The conjugate of √3+√6 is √3 - √6, so let's do it:

\frac{ \sqrt{3} - \sqrt{6} }{ \sqrt{3} + \sqrt{6} } . \frac{ \sqrt{3}- \sqrt{6}  }{ \sqrt{3}- \sqrt{6} }

To help you with the solution of that expression, I will show each part.

1) Numerator: (√3 - √6) . (√3 - √6) = (√3 - √6)^2 = (√3)^2 - 2√3√6 + (√6)^2 =

= 3 - 2√18 + 6 = 9 - 6√2.

2) Denominator: (√3 + √6).(√3 - √6) = (√3)^2 - (√6)^2 = 3 - 6 = - 3

3) Then the resulting expression is:

 9 - 6√2
-----------
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Which can be further simplified, dividing by - 3

    -3 + 2√2

Answer: 2√2 - 3

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