Answer:
y(t) = 2.5 e⁶ᵗ + 2.5 e⁻⁶ᵗ
Or
y(t) = 5 e⁻⁶ᵗ
Step-by-step explanation:
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ
Let us find our value for y(t) that satisfies the conditions
1) y" - 36y = 0
y" = (d²y/dt²)
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ
y' = (dy/dt) = 6c₁ e⁶ᵗ - 6c₂ e⁻⁶ᵗ
y" = (d/dt)(dy/dt) = 36c₁ e⁶ᵗ + 36c₂ e⁻⁶ᵗ
y" - 36y = 36c₁ e⁶ᵗ + 36c₂ e⁻⁶ᵗ - 36(c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ) = 36c₁ e⁶ᵗ + 36c₂ e⁻⁶ᵗ - 36c₁ e⁶ᵗ - 36c₂ e⁻⁶ᵗ = 0.
The function satisfies this condition.
2) y(0) = 5
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ
At t = 0
y(0) = c₁ e⁰ + c₂ e⁰ = 5
c₁ + c₂ = 5 (e⁰ = 1)
3) lim t→+[infinity] y(t)=0
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ = 0 as t→+[infinity]
c₁ e⁶ᵗ = - c₂ e⁻⁶ᵗ as t→+[infinity]
c₁ = - c₂ e⁻¹²ᵗ as t→+[infinity]
e⁻¹²ᵗ = 0 as t→+[infinity]
c₁ = c₂ or c₁ = 0
Recall c₁ + c₂ = 5
If c₁ = 0, c₂ = 5
If c₁ = c₂, c₁ = c₂ = 2.5
y(t) = c₁ e⁶ᵗ + c₂ e⁻⁶ᵗ = 2.5 e⁶ᵗ + 2.5 e⁻⁶ᵗ
Or
y(t) = 5 e⁻⁶ᵗ
Answer:
x = 30/7
Step-by-step explanation:
2/3x = − 1/2x + 5
multiply by 6 to clear the fractions
4x = -3x + 30
add 3x to both sides
7x = 30
Divide both sides by 7
x = 30/7
The mathematical model representing the scenarios described can be expressed as follows :
- If n ≤ 8 ; A(n) = 31.25n
- If n > 8 ; A(n) = 31.25 + 46.875(n - 8)
1.)
Amount paid per 8 hours at summer job = 250
Additional hours beyond, 8 hours = 1.5 × hourly rate
Let:
- hourly rate = p
- Number of hours worked = n
- Amount earned as a function of n, A(n)
Hourly rate, p = (250 ÷ 8) = 31.25
Therefore, the hourly rate, p at the summer job = $31.25
Overtime pay = 1.5 × 31.25 = $46.875
Therefore,
If n ≤ 8 ; A(n) = 31.25n
If n > 8 ; A(n) = 31.25 + 46.875(n - 8)
2.)
Let :
- Number of playing hours = n
- Total amount charged as a function of n ; T(n)
Therefore,
T(n) = First hour fee + (charge per additional hour × number of additional hours)
T(n) = 15 + 5(n - 1)
Therefore, the models can be used to calculate the total earning and amount charged for any given hour value.
Learn more :brainly.com/question/18112348