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Answer :
Let's tackle each part of the question step by step:
a. How many weeks will it take for the number of plants to double?
To find out how many weeks it will take for the number of plants to double, we need to determine when the number of plants is twice the initial amount.
We start with 10 plants initially. If the number doubles, we want the number of plants to be 20. The function modeling the plant growth after [tex]\( w \)[/tex] weeks is:
[tex]\[ p(w) = 10 \times 2^w \][/tex]
We set this equation equal to 20 to find the doubling time:
[tex]\[ 20 = 10 \times 2^w \][/tex]
Divide both sides by 10:
[tex]\[ 2 = 2^w \][/tex]
This implies:
[tex]\[ 2^1 = 2^w \][/tex]
Therefore, [tex]\( w = 1 \)[/tex].
It will take 1 week for the number of plants to double.
b. How many times will the initial number of plants double during a 12-week period?
Since the plants double every 1 week, within a 12-week period, the number of times the plants will double is simply the number of weeks divided by the doubling time:
[tex]\[ \text{times doubled} = \frac{12 \text{ weeks}}{1 \text{ week per doubling}} = 12 \][/tex]
So, during a 12-week period, the number of plants will double 12 times.
c. Rewrite the equation of the function p using [tex]\( d \)[/tex] days as the input instead of [tex]\( w \)[/tex] weeks.
Since a week consists of 7 days, we need to adjust the growth model from weeks to days. If [tex]\( d \)[/tex] is the number of days, then:
[tex]\[ w = \frac{d}{7} \][/tex]
Substitute [tex]\( w = \frac{d}{7} \)[/tex] into the original equation:
[tex]\[ p(w) = 10 \times 2^w \][/tex]
[tex]\[ p(d) = 10 \times 2^{\frac{d}{7}} \][/tex]
Thus, the revised equation is:
[tex]\[ p(d) = 10 \times 2^{\frac{d}{7}} \][/tex]
This new equation represents the number of water hyacinth plants in the pond after [tex]\( d \)[/tex] days.
a. How many weeks will it take for the number of plants to double?
To find out how many weeks it will take for the number of plants to double, we need to determine when the number of plants is twice the initial amount.
We start with 10 plants initially. If the number doubles, we want the number of plants to be 20. The function modeling the plant growth after [tex]\( w \)[/tex] weeks is:
[tex]\[ p(w) = 10 \times 2^w \][/tex]
We set this equation equal to 20 to find the doubling time:
[tex]\[ 20 = 10 \times 2^w \][/tex]
Divide both sides by 10:
[tex]\[ 2 = 2^w \][/tex]
This implies:
[tex]\[ 2^1 = 2^w \][/tex]
Therefore, [tex]\( w = 1 \)[/tex].
It will take 1 week for the number of plants to double.
b. How many times will the initial number of plants double during a 12-week period?
Since the plants double every 1 week, within a 12-week period, the number of times the plants will double is simply the number of weeks divided by the doubling time:
[tex]\[ \text{times doubled} = \frac{12 \text{ weeks}}{1 \text{ week per doubling}} = 12 \][/tex]
So, during a 12-week period, the number of plants will double 12 times.
c. Rewrite the equation of the function p using [tex]\( d \)[/tex] days as the input instead of [tex]\( w \)[/tex] weeks.
Since a week consists of 7 days, we need to adjust the growth model from weeks to days. If [tex]\( d \)[/tex] is the number of days, then:
[tex]\[ w = \frac{d}{7} \][/tex]
Substitute [tex]\( w = \frac{d}{7} \)[/tex] into the original equation:
[tex]\[ p(w) = 10 \times 2^w \][/tex]
[tex]\[ p(d) = 10 \times 2^{\frac{d}{7}} \][/tex]
Thus, the revised equation is:
[tex]\[ p(d) = 10 \times 2^{\frac{d}{7}} \][/tex]
This new equation represents the number of water hyacinth plants in the pond after [tex]\( d \)[/tex] days.
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