Harold and Millicent are getting married and need to combine their already-full libraries. If Harold, who has 1/2 as many books
1 answer:
Answer:
2/3
Step-by-step explanation:
Let x be the number of books that Millicent has. So the number of books that Harold has is x/2. And the number of book that Harold is bringing to the new home is (x/2)/3 = x/6.
The home has enough room for another half of Millicent's books, which is x/2. So the total books capacity of the new home is
Since x is also the number of books in Millicent's old library capacity, we can conclude that the fraction of Millicent's old library capacity to the new home's library capacity is 2/3, or 2 thirds
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Let z = 3x+2. Then, by substitution, your equation is
.. z^2 +7z -8 = 0
_____
This equation can be factored as
.. (z +8)(z -1) = 0
.. x = -8 or +1
Then
.. 3x +2 = -8 or +1
.. 3x = -10 or -1
.. x = -10/3 or -1/3
.. z^2 +7z -8 = 0
_____
This equation can be factored as
.. (z +8)(z -1) = 0
.. x = -8 or +1
Then
.. 3x +2 = -8 or +1
.. 3x = -10 or -1
.. x = -10/3 or -1/3
Answer:
(a)
(b) (i) The population after 8 hours is 607,325
(ii) The population after 24 hours is 1,828,643
(c) The rate of increase of the population as a percentage per hour is 7.132%
(d) The doubling time of the population is approximately, 10.06 hours
Step-by-step explanation:
(a) The initial population of the bacteria, y₁ = a = 350,000
The time the colony grows, t = 12 hours
The final population of bacteria in the colony, y₂ = 800,000
The exponential growth model, can be written as follows;
Plugging in the values, we get;
Therefore;
(1 + r)¹² = 800,000/350,000 = 16/7
12·㏑(1 + r) = ㏑(16/7)
㏑(1 + r) = (㏑(16/7))/12
r = e^((㏑(16/7))/12) - 1 ≈ 0.07132
The model is therefore;
(b) (i) The population after 8 hours is given as follows;
y = 350,000 × (1 + 0.07132)⁸ ≈ 607,325.82
By rounding down, we have;
The population after 8 hours, y = 607,325
(ii) The population after 24 hours is given as follows;
y = 350,000 × (1 + 0.07132)²⁴ ≈ 1,828,643.92571
By rounding down, we have;
The population after 24 hours, y = 1,828,643
(c) The rate of increase of the population as a percentage per hour = r × 100
∴ The rate of increase of the population as a percentage = 0.07132 × 100 = 7.132%
(d) The doubling time of the population is the time it takes the population to double, which is given as follows;
Initial population = y
Final population = 2·y
The doubling time of the population is therefore;
Therefore, we have;
2·y/y =2 =
t = ln2/(ln(1 + 0.07132)) ≈ 10.06
The doubling time of the population is approximately, 10.06 hours.