An actor invested some money at 6% simple interest, and $48 comma 000 more than three times the amount at 7 %. the total annual interest earned from the investment was $37 comma 110. how much did he invest at each amount?
he invested $__
nothing at 6% and __
nothing at 7%

He invested $125,000 at 6% and $423,000 at 7%.

Step-by-step explanation:

Let be the amount of money invested at 6%. Since the amount of money invested at 7% is 3 times the amount of money invested at 6% plus $48,000, is the amount invested at 7%.

We know that both investments yield $37,110 in interest, so the sum of the money invested at 6% and the money invested at 7% is $37,110; in other words:

6%+7% = 37,110

We need to convert both interest rates to decimals, so we are going to divide both rates by 100%

6%+7% = 37,110

Now we can solve our equation, step-by-step, to find the amount of each investment.

Step 1. Distribute 0.07 to 3x and 48,000

Step 2. Combine like terms

Step 3. Subtract 3,360 from both sides of the equation

Step 4. Divide both sides of the equation by 0.27

We now know that he invested $125,000 at 6%. Since the amount invested at 7% is , we just need to replace x with 125,000 to find it:

Amount invested at 7% =

Amount invested at 7% =

Amount invested at 7% =

We can conclude that the actor invested $125,000 at 6% and $423,000 at 7%.

conservation of angular momentum ; conservation of energy

Step-by-step explanation:

The complete Question is given as follows:

" A uniform bar of mass (m) and length (L) is suspended on a frictionless hinge. A horizontally launched blob of clay of mass (m) strikes the bottom end of the bar and sticks to it. After that, the bar swings upward. What is the minimum initial speed (v) of the blob of clay that would enable the rod to swing a full circle? Which concepts/laws would be most helpful in solving this problem? Select the best answer from the options below.  "

CHOICES

kinematics of rotational motion; conservation of energy

conservation of momentum ; conservation of energy

conservation of angular momentum ; conservation of momentum

conservation of angular momentum ; conservation of energy

conservation of energy ; Newton's laws

Newton's laws ; conservation of angular momentum

conservation of angular momentum ; kinematics of rotational motion

Newton's laws, kinematics of rotational motion

Solution:

- We will apply the conservation of angular momentum M. Note the linear momentum does not remains conserved as the rod stores some energy as the clay sticks to the rod:

                                       M_i = M_f

- Initially the rod was at rest and clay had velocity of v, then M_i can be written as:

                                       M_i = m*v*L

- The final momentum is the combined effect of clay and rod:

                                       M_f = ( m*L^2 + I_rod )*w

- Where w is the angular speed of the rod after impact. And I_rod is the moment of inertia of rod.

                                      I_rod = mL^2 / 3

                                      M_f = ( m*L^2 + m*L^2 /3 )*w = (4*m*L^2 / 3)*w

- Formulate w in terms of initial velocity v:

                                     m*v*L = (4*m*L^2 / 3)*w

                                      0.75*v / L = w

- The minimum amount of velocity required would be enough to complete half of a circle.

- Apply conservation of Energy principle:

                                     T_i + V_i = T_f + V_f

Where, T is the kinetic energy soon after impact and at top most position.

            Assuming, T_f = 0 , for minimum velocity required to complete on circle.

             T_i = 0.5*I_combined*w^2

Where, I_combined = I_clay + I_rod = 4*m*L^2 / 3

And w = 0.75*v / L:

             T_i = 0.5*[4*m*L^2 / 3]*[0.75*v / L]^2

             T_i = 0.5*[m]*[v^2]

Also, V is the potential energy of the clay plus rod system soon after impact and at top most point.

             V_i = 0 ( Datum )

             V_f = V_rod + V_clay

             V_f = m*g*L + m*g*2L = 3*m*g*L

- Plug in the expressions in the energy balance and we get:

                              0.5*[m]*[v^2] + 0 = 0 + 3*m*g*L

                                      v_min = sqrt ( 6*g*L)

- So the choices used were:

conservation of angular momentum ; conservation of energy