Sales at a certain department store follow the model 4072-01-02-03-00_files/i0100000.jpg where y is the total sales in thousands of dollars and x is the number of years after 2001. What was the first year that sales fell below $50,000?
To find the first year that sales fell below $50,000, we need to solve the equation y < 50, where y represents the total sales in thousands of dollars.
According to the given model 4072-01-02-03-00_files/i0100000.jpg, y is a function of x, where x represents the number of years after 2001. So, we can rewrite the equation as:
4072-01-02-03-00_files/i0100001.jpg < 50
Now, let's substitute the expression for y from the model:
4072-01-02-03-00_files/i0100001.jpg < 50
Simplify the equation:
(4072 - 2x - 3x^2) < 50
Rearrange the equation:
3x^2 + 2x - 4022 > 0
Now, we need to solve this quadratic inequality to find the range of x values when sales fall below $50,000. We can use different methods like factoring, completing the square, or using the quadratic formula. But let's solve it using the quadratic formula:
The quadratic formula is given as:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic inequality, the values of a, b, and c are:
a = 3
b = 2
c = -4022
Substitute these values into the quadratic formula:
x = (-2 ± √(2^2 - 4 * 3 * -4022)) / (2 * 3)
Simplify the expression under the square root:
x = (-2 ± √(4 + 24132)) / 6
x = (-2 ± √24136) / 6
x = (-2 ± 155.4) / 6
Now, we have two possible solutions:
1) x = (-2 + 155.4) / 6
2) x = (-2 - 155.4) / 6
2) gives a negative value, which doesn't make sense in this context. So, we can ignore it.
Solve for x in 1):
x = (153.4) / 6
x ≈ 25.57
Since x represents the number of years after 2001, we need to round up this value to the nearest whole number. Therefore, the first year that sales fell below $50,000 is 26 years after 2001.
Adding 2001, we get:
2001 + 26 = 2027
So, the first year that sales fell below $50,000 is 2027.
To find the first year that sales fell below $50,000, we need to set up the equation and solve for x.
Given:
Sales model: y = 4072(0.01)^x(0.02)^x(0.03)^x(0.00)^x, where y is the total sales in thousands of dollars and x is the number of years after 2001.
We have y = 50 (since sales fell below $50,000).
Substituting the given values, we get:
50 = 4072(0.01)^x(0.02)^x(0.03)^x(0.00)^x
Let's simplify the equation step-by-step:
Step 1: Simplify the expression. Note that any number raised to the power of 0 is equal to 1.
50 = 4072(1)(0.01)^x(0.02)^x(0.03)^x(0.00)^x
Step 2: Combine the multiplication.
50 = 4072(0.01)^x(0.02)^x(0.03)^x(0.00)^x
Step 3: To make it simpler, we can ignore the multiplication by 0.
50 = 4072(0.01)^x(0.02)^x(0.03)^x
Step 4: Since all the factors are less than 1, we can rewrite the equation in exponential form.
50 = 4072(1/100)^x(2/100)^x(3/100)^x
Step 5: Simplify the powers of fractions.
50 = 4072(1/100)^x(2/100)^x(3/100)^x
50 = 4072(1/100*x)(1/50*x)(3/100)^x
Step 6: Combine the fractions with the same base.
50 = 4072(3/50/100)^x
Step 7: Simplify the expression in parentheses.
50 = 4072(3/5000)^x
Step 8: Divide both sides by 4072.
50/4072 = (3/5000)^x
Step 9: Calculate the left side of the equation.
0.0123 = (3/5000)^x
Step 10: Take the logarithm of both sides to solve for x. We can use either natural logarithm (ln) or logarithm with base 10 (log).
log(0.0123) = log[(3/5000)^x]
Step 11: Simplify the equation.
x * log(3/5000) = log(0.0123)
Step 12: Divide both sides by log(3/5000).
x = log(0.0123) / log(3/5000)
Using a calculator, we can evaluate the right side of the equation.
Step 13: Calculate x using a calculator.
x ≈ 13.02
Step 14: Since x represents the number of years after 2001, we add x to 2001 to find the first year that sales fell below $50,000.
First year = 2001 + 13.02
Step 15: Calculate the first year.
First year ≈ 2014
Therefore, the first year that sales fell below $50,000 was approximately 2014.