Can someone check my answers for the following questions?
1) During the period 2010-2018 per capita consumption of honey increased from 14 ounces in 2010 (x=0) by 0.56 ounces per year.
a) Find a linear function, f(x) giving the per capita consumption of honey in the year x, where x=0 corresponds to the beginning of 2010.
b) If this trend continues, what will be the per capita consumption of honey in the beginning of 2025?
a) 14+0.46=14.56
f(0) = a0+14=14.56 i don't know if it is set up right
b) f(15) =15a+14=14.56
2) The percentage of high school students who drink and drive was 14.5% at the beginning of 2004 and declined linearly to 9.8% at the beginning of 2014.
a) Find a linear ft giving the percentage of high school students who drink and drive in year t where t=0 corresponds to the beginning of 2004.
b) If this trend continues, what will the percentage of high school students who will drink and drive be at the beginning of 2020?
a)f(0) = a0+b=14.5
2014-2004=10
f(10) =10a+14.5%=9.8
b)2020-2004=16
f(16)=16a+14.5%=9.8
3) Find the equation of the line passing through the point (a,b) with slope equal to zero.
y=0x+b
4) a) Find the vertical line through the point (2,5)
b) Find the y-intercept if the line y=9.
a) x=2
b) b=9
#1a) almost. You need a variable x. You have implicitly assigned x=1. Try
f(x) = 14+0.46x
b) f(15) = 14 + 0.46*15 = 20.9
weren't you suspicious when (b) was the same value as (a) ?
#2 a) The % changed by 9.8-14.5 = -4.7 in 10 years.
That is a -0.47% change each year. So,
f(x) = 14.5 - 0.47x
b) so now find f(16) as above
#3 correct, but most people would just write it as
y = b
#4 correct, but you don't really need to use b.
the y-intercept is the point (0,9)
or, since we assume the y-axis, just y=9
Oh. Above was when I was trying to make sense of the way they taught me but your way makes more sense. Thank you.
Let's go through each question and check your answers while explaining the steps to get the correct answers.
1) The given information states that during the period 2010-2018, per capita consumption of honey increased from 14 ounces in 2010 (x=0) by 0.56 ounces per year.
a) To find a linear function, f(x), giving the per capita consumption of honey in the year x, where x=0 corresponds to the beginning of 2010, you can use the formula for a linear equation: f(x) = a * x + b.
In this case, we need to determine the values of 'a' and 'b' to construct the linear function. Given that the consumption increased by 0.56 ounces per year, we can determine the value of 'a' as follows:
a * 8 = 0.56 (since the period is from 2010 to 2018, which is 8 years)
a = 0.56 / 8
a = 0.07
Now, to find 'b', we can substitute the values of x and f(x) for a given point. We know that the consumption in the beginning of 2010 (x=0) was 14 ounces, so we can set up the equation:
14 = a * 0 + b
14 = b
Therefore, the linear function f(x) is:
f(x) = 0.07 * x + 14
b) To find the per capita consumption of honey in the beginning of 2025, we need to evaluate f(x) for x = 15 (since 2025 - 2010 = 15).
f(15) = 0.07 * 15 + 14
f(15) = 1.05 + 14
f(15) = 15.05
Therefore, the per capita consumption of honey in the beginning of 2025 will be approximately 15.05 ounces.
2) The given information states that the percentage of high school students who drink and drive was 14.5% at the beginning of 2004 and declined linearly to 9.8% at the beginning of 2014.
a) To find a linear function, f(t), giving the percentage of high school students who drink and drive in year t, where t=0 corresponds to the beginning of 2004, we can use the linear equation formula: f(t) = a * t + b.
In this case, we need to determine the values of 'a' and 'b'. The given information tells us that the percentage declined linearly over a period of 10 years, from 2004 to 2014. So the change in percentage over this period can be calculated as:
change in percentage = 14.5% - 9.8% = 4.7%
Now we can calculate the value of 'a' using the formula: a = (change in percentage) / (years).
a = (4.7%) / (10 years)
a = 0.47%
To find 'b', we can substitute the values of t and f(t) for a given point. We know that at the beginning of 2004 (t=0), the percentage was 14.5%, so:
14.5% = a * 0 + b
14.5% = b
Therefore, the linear function f(t) is:
f(t) = 0.47% * t + 14.5%
b) To find the percentage of high school students who will drink and drive at the beginning of 2020, we need to evaluate f(t) for t = 16 (since 2020 - 2004 = 16).
f(16) = 0.47% * 16 + 14.5%
f(16) = 7.52% + 14.5%
f(16) = 22.02%
Therefore, the percentage of high school students who will drink and drive at the beginning of 2020 will be approximately 22.02%.
3) To find the equation of the line passing through the point (a, b) with a slope equal to zero, we know that a zero slope means the line is horizontal.
Therefore, the equation of the line passing through the point (a, b) with slope zero is:
y = b
This is because a horizontal line has the same y-coordinate value for any x-coordinate.
4)
a) To find the vertical line through the point (2, 5), we know that a vertical line has a constant x-coordinate value. In this case, since the point is (2, 5), the equation of the vertical line is:
x = 2
This means that for any value of y, the x-coordinate will always be 2.
b) To find the y-intercept if the line y = 9, we need to remember that the y-intercept is the point where the line intersects the y-axis. In this case, the equation of the line is:
y = 9
The y-intercept is the value of y when x is zero. Therefore, at x = 0, the equation becomes:
y = 9
So the y-intercept is 9.