For 1985 through 1996, the number, C (in thousands), of videos rented each year in Moose Jaw can be modeled by C= 0.069(t^3+4t^2+37t+600) where t=0 represents 1990. During which year are 60.4 thousand movies projected to be rented?
Could someone please show me the steps to this problem?
You have the equation. Just solve for t:
C= 0.069(t^3+4t^2+37t+600)
60.4 = 0.069(t^3+4t^2+37t+600)
875.36 = t^3+4t^2+37t+600
t^3 + 4t^2 + 37t - 275.36 = 0
Cubics are hard to solve, but there are many online polynomial solvers. We get
t = 3.99, or just 4.
So, it looks like 1994 is the year you want.
Note: 4 is an exact root of
t^3 + 4t^2 + 37t - 276 = 0
which could have been found using trials of synthetic division.
To find the year during which 60.4 thousand movies are projected to be rented, we need to solve the equation C = 60.4.
Given:
C = 0.069(t^3 + 4t^2 + 37t + 600)
Substitute C with 60.4:
60.4 = 0.069(t^3 + 4t^2 + 37t + 600)
Next, we will solve this equation to find the value of t.
1. Divide both sides of the equation by 0.069:
60.4/0.069 = t^3 + 4t^2 + 37t + 600
2. Simplify the equation:
876.81 = t^3 + 4t^2 + 37t + 600
3. Rearrange the equation to the standard form of a polynomial equation:
t^3 + 4t^2 + 37t + 600 - 876.81 = 0
4. Combine like terms:
t^3 + 4t^2 + 37t - 276.81 = 0
Now, we need to find the value of t, which represents the year. This equation can be solved using various methods such as factoring, completing the square, or the cubic formula. Since the equation is a cubic equation, we will use the cubic formula to find the roots.
The cubic formula is given by:
t = (-b ± √(b^2 - 4ac))/(2a) - (1/3)(A/(t + B))
In this equation:
a = 1, b = 4, c = 37, and d = -276.81
5. Substitute the values into the cubic formula:
t = (-4 ± √(4^2 - 4(1)(37)(-276.81)))/(2(1)) - (1/3)(-276.81/(t + ((√(37^2 - 3(4^3))/(3(1))))^(1/3)))
6. Simplify the equation:
t = (-4 ± √(16 + 31058.64))/(2) - (1/3)(-276.81/(t + ((√(4453))/(3))))
7. Solve the inner square root:
t = (-4 ± √(31074.64))/(2) - (1/3)(-276.81/(t + ((66.775)/(3))))
8. Simplify the square root:
t = (-4 ± 176.205)/(2) - (1/3)(-276.81/(t + 22.2583))
9. Simplify further:
t = (-4 ± 176.205)/(2) - (-92.2705/(t + 22.2583))
t = (-4 ± 176.205)/(2) + 92.2705/(t + 22.2583)
10. Solve for t using the positive root:
t = (-4 + 176.205)/(2) + 92.2705/(t + 22.2583)
11. Simplify further:
t = (172.205)/(2) + 92.2705/(t + 22.2583)
12. Simplify and solve for t:
t = 86.1025 + 92.2705/(t + 22.2583)
13. Rearrange the equation:
t - 86.1025 = 92.2705/(t + 22.2583)
14. Multiply both sides by (t + 22.2583):
t(t + 22.2583) - 86.1025(t + 22.2583) = 92.2705
15. Distribute and simplify:
t^2 + 22.2583t - 86.1025t - 1925.94 = 92.2705
16. Combine like terms:
t^2 - 63.8442t - 2018.2105 = 0
Now, we can use the quadratic formula to solve for t:
t = (-(-63.8442) ± √((-63.8442)^2 - 4(1)(-2018.2105)))/(2(1))
17. Simplify the equation:
t = (63.8442 ± √(4076.57728164 + 8072.841)) / 2
t = (63.8442 ± √(4076.57728164 + 8072.841)) / 2
t = (63.8442 ± √(12149.41828164)) / 2
t = (63.8442 ± 110.155657) / 2
18. Solve for t using the positive root:
t = (63.8442 + 110.155657) / 2
t = 173.999857 / 2
t ≈ 87
Therefore, 60.4 thousand movies are projected to be rented approximately in the year 1987.
Sure! To find the year during which 60.4 thousand movies are projected to be rented, we need to substitute C with 60.4 in the given equation and solve for t (representing the year).
The equation given is:
C= 0.069(t^3+4t^2+37t+600)
Substitute C with 60.4:
60.4 = 0.069(t^3+4t^2+37t+600)
Now, let's solve for t. Start by subtracting 60.4 from both sides of the equation:
0 = 0.069(t^3+4t^2+37t+600) - 60.4
0 = 0.069(t^3+4t^2+37t+600) - 60.4
Next, you can divide both sides by 0.069 to isolate the expression in the parentheses:
0 / 0.069 = t^3+4t^2+37t+600 - 60.4 / 0.069
0 = t^3 + 4t^2 + 37t + 600 - 876.81
Simplify further:
0 = t^3 + 4t^2 + 37t - 276.81
Now, to find the value of t, we need to solve this cubic equation. Since solving cubic equations can be complex, one way to approach this is by using numerical methods or using a graphing calculator.
We can use a graphing calculator to find the roots of this equation. Graph the equation y = t^3 + 4t^2 + 37t - 276.81, and find the x-intercept where y = 0.
Once you have the x-intercept (the t-value), you can add it to 1990 (since t=0 represents 1990) to find the year during which 60.4 thousand movies are projected to be rented.
I hope this explanation helps! Let me know if you have any more questions.