signment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change ent Scoring submission is used for your score. Points] DETAILS MY NOTES SPRECALC8 \( \mathbf{1 . 7 . 0 2 6} \). ASK YOUR an amount of \( \$ 7,000 \) is invested at a simple interest rate of \( 4 \frac{1}{2} \% \) interest per year, how much additional money (in dollars) must be invested at a simple interest \( r \) ensure that the interest each year is \( 5 \% \) of the total amount invested? \( \$ \square \) Need Help? Read li

To determine the additional amount of money that needs to be invested, let's break down the problem step by step. ### Given: - **Initial Investment (P₁):** \$7,000 - **Interest Rate on Initial Investment (r₁):** 4.5% per year - **Desired Total Interest Rate (R):** 5% per year - **Additional Investment (P₂):** ? - **Interest Rate on Additional Investment (r₂):** \( r \) per year ### Objective: Find the additional investment \( P₂ \) such that the total interest from both investments equals 5% of the total amount invested. ### Steps: 1. **Calculate the Interest from the Initial Investment:** \[ \text{Interest from } P₁ = P₁ \times r₁ = 7000 \times 0.045 = \$315 \] 2. **Define the Total Investment and Desired Interest:** \[ \text{Total Investment} = P₁ + P₂ = 7000 + P₂ \] \[ \text{Total Desired Interest} = \text{Total Investment} \times R = (7000 + P₂) \times 0.05 \] 3. **Set Up the Equation for Total Interest:** \[ \text{Interest from } P₁ + \text{Interest from } P₂ = \text{Total Desired Interest} \] \[ 315 + P₂ \times r = 0.05 \times (7000 + P₂) \] 4. **Solve for \( P₂ \):** \[ 315 + P₂ \times r = 350 + 0.05P₂ \] \[ P₂ \times r - 0.05P₂ = 350 - 315 \] \[ P₂ (r - 0.05) = 35 \] \[ P₂ = \frac{35}{r - 0.05} \] ### Final Answer: The additional amount to be invested is: \[ \boxed{\,\,P₂ = \dfrac{35}{\,r\;-\;0.05\,} \,\,$\,\,} \] ### Example: If the additional investment rate \( r \) is 7% (or 0.07 in decimal): \[ P₂ = \frac{35}{0.07 - 0.05} = \frac{35}{0.02} = \$1,750 \] So, you would need to invest an additional **\$1,750** at a 7% interest rate to achieve the desired total interest rate of 5%.