The minimum hourly wage,
y
(in dollars per hour), in a country can be approximated by the equation
=y+0.15x2.92
. In this equation,
x
represents the number of years since
1970
=x=0 represents 1970, x5 represents 1975, and so on
.
Federal Minimum Hourly Wage by Year
(a) Use the equation to approximate the minimum wage in the year
1995
.
The minimum wage in
1995
was approximately
$
.
(b) Use the equation to estimate the minimum wage in the year
2005
.
The minimum wage in
2005
was approximately
$
.
(c) Determine the
y
-intercept. Interpret the meaning of the
y
-intercept in the context of this problem.
The
y
-intercept is
,
. In the year
the minimum wage was approximately
$
per hour.
(d) Determine the slope. Interpret the meaning of the slope in the context of this problem.
The slope is
. This indicates that the minimum wage has (increased or decreased) by approximately
$
per year during this period.
(a) To find the minimum wage in 1995, we substitute x=1995-1970=25 into the equation:
y = 0.15x^2.92
y = 0.15(25)^2.92
y ≈ 0.15(656.1)
y ≈ 98.415
The minimum wage in 1995 was approximately $98.42.
(b) To estimate the minimum wage in 2005, we substitute x=2005-1970=35 into the equation:
y = 0.15x^2.92
y = 0.15(35)^2.92
y ≈ 0.15(2252.4854)
y ≈ 337.873
The minimum wage in 2005 was approximately $337.87.
(c) The y-intercept is the value of y when x=0. Substituting x=0 into the equation:
y = 0.15x^2.92
y = 0.15(0)^2.92
y = 0
The y-intercept is 0. In the context of this problem, it means that in the year 1970, the minimum wage was approximately $0 per hour. This could be because there was no federal minimum wage in place at that time.
(d) The slope represents the rate at which the minimum wage is changing over time. In this case, the slope is 0.15. This indicates that the minimum wage has increased by approximately $0.15 per year during this period.
(a) To approximate the minimum wage in the year 1995, we need to substitute the value of x = 1995 - 1970 = 25 into the equation y = 0.15x^2.92.
y = 0.15(25)^2.92
y ≈ 0.15(610) [Using a calculator to evaluate 25^2.92]
y ≈ 91.5
Therefore, the minimum wage in 1995 was approximately $91.50 per hour.
(b) To estimate the minimum wage in the year 2005, we need to substitute the value of x = 2005 - 1970 = 35 into the equation y = 0.15x^2.92.
y = 0.15(35)^2.92
y ≈ 0.15(1,584) [Using a calculator to evaluate 35^2.92]
y ≈ 237.6
Therefore, the minimum wage in 2005 was approximately $237.60 per hour.
(c) The y-intercept is the value of y when x = 0. Substitute x = 0 into the equation y = 0.15x^2.92.
y = 0.15(0)^2.92
y ≈ 0
The y-intercept is (0, 0). In the context of this problem, it means that in the year 1970, the minimum wage was approximately $0 per hour. However, it's important to note that this is just an approximation and might not reflect the actual minimum wage during that time.
(d) The slope of the equation y = 0.15x^2.92 represents the rate at which the minimum wage changes per year. In this case, the slope is 0.15.
Interpreting the meaning of the slope, it indicates that the minimum wage has increased by approximately $0.15 per year during this period.
(a) To approximate the minimum wage in the year 1995, we substitute x = 1995 - 1970 = 25 into the equation.
y = y + 0.15x^2.92
y = y + 0.15(25)^2.92
Using a calculator, we can evaluate this expression:
y ≈ y + 0.15(25)^2.92
y ≈ y + 0.15(259.2673)
y ≈ y + 38.89
Therefore, the minimum wage in 1995 was approximately $39 per hour.
(b) To estimate the minimum wage in the year 2005, we substitute x = 2005 - 1970 = 35 into the equation.
y = y + 0.15x^2.92
y = y + 0.15(35)^2.92
Using a calculator, we can evaluate this expression:
y ≈ y + 0.15(35)^2.92
y ≈ y + 0.15(837.3959)
y ≈ y + 125.61
Therefore, the minimum wage in 2005 was approximately $126 per hour.
(c) The y-intercept refers to the value of y when x = 0. In the context of this problem, when x = 0 corresponds to the year 1970.
So, the y-intercept represents the minimum wage in 1970.
The y-intercept is y = 0 + 0.15(0)^2.92 = $0.
Therefore, in 1970, the minimum wage was approximately $0 per hour.
(d) The slope of the equation indicates how the minimum wage changes as time (represented by x) increases.
The slope is 0.15.
Interpreting this in the context of the problem, the slope of 0.15 means that the minimum wage has increased by approximately $0.15 per year during this period.
Please note that in the original equation provided, there seems to be a missing variable for y, making it difficult to provide precise calculations. However, the steps above outline the general process for solving these types of problems using equations.