An inheritance of $\$24,000$ is to be divided among three trusts, with the second trust receiving twice as much as the first trust. The three trusts pay interest at the rates of $9\%$, $10\%$, and $6\%$ annually respectively, and return a total in interest of $\$2210$ at the end of the first year. How much was invested in each trust?

The force F has a magnitude of 800 N. Express F as a vector in terms of the unit vectors i and j. Identify the x and y scalar components of F .

For the given vectors $V_1$ and $V_2$ of Prob. 1/2, determine the magnitude of the vector difference $V' = V_2 - V_1$ and the angle $\theta_x$ which $V'$ makes with the positive x-axis. Complete both graphical and algebraic solutions.

Solve the linear system with the given augmented matrix.

(a)

$$\left[\begin{array}{lll:l}1 & 2 & 3 & 0 \\ 1 & 1 & 1 & 0 \\ 5 & 7 & 9 & 0\end{array}\right]$$

(b)

$$\left[\begin{array}{rrr:r}1 & 2 & 1 & 7 \\ 2 & 0 & 1 & 4 \\ 1 & 0 & 2 & 5 \\ 1 & 2 & 3 & 11 \\ 2 & 1 & 4 & 12\end{array}\right]$$

A woman at a point A on the shore of a circular lake with radius 2 mi wants to arrive at the point C diametrically opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 4 mi/h and row a boat at 2 mi/h. How should she proceed?.

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

$$\begin{array}{|c|c|} \hline \text{t(min)} & \text{Heartbeats}\\ \hline \text{36} & \text{2530}\\ \text{38} & \text{2661}\\ \text{40} & \text{2806}\\ \text{42} & \text{2948}\\ \text{44} & \text{3080}\\ \hline \end{array}$$

(a) Find the average heart rates (slopes of the secant lines) over the time intervals [40, 42] and [42, 44]. (b) Estimate the patient’s heart rate after 42 min by averaging the slopes of these two secant lines.

If

$$a) A = \begin{bmatrix} a+b & c+d \\ c-d& a-b \\ \end{bmatrix}= \begin{bmatrix} 4 & 6 \\ 10 &12 \\ \end{bmatrix}$$

find a, b and c.

Show that if $A$ is an $n \times n$ matrix, then $A$ can be written uniquely as $A=S+K$, where $S$ is symmetric and $K$ is skew symmetric.

If A is a nonsingular matrix whose inverse is

$$\begin{bmatrix}2 & 1 \\ 4 & 1\end{bmatrix}$$

, find A.

Write the linear system whos argumented matrix is
a)

$$\begin{bmatrix} 2& 1 & 3& 4& \text{\textbar} & 0\\ 3& -1 & 2& 0& \text{\textbar} & 3\\ -2& 1 & -4& 3& \text{\textbar} & 2\\ \end{bmatrix}$$

b)

$$\begin{bmatrix} 2& 1 & 3& 4& \text{\textbar} & 0\\ 3& -1 & 2& 0& \text{\textbar} & 3\\ -2& 1 & -4& 3& \text{\textbar} & 2\\ 0& 0 & 0& 0& \text{\textbar} & 0\\ \end{bmatrix}$$

A force is specified by the vector F = 160i + 80j 120k N. Calculate the angles made by F with the positive x-, y-, and z-axes.

Determine the angles made by the vector V = - 36i + 15j with the positive x- and y-axes. Write the unit vector n in the direction of V.

Suppose that the three points (1,-5). (- 1. 1). and (2. 7) lie on the parabola $p(x)=ax^2+bx+c$.

(a) Determine a linear system of three equations in three unknowns that must be solved to find a, b and c.

(b) Solve the linear system obtained in part (a) for a, b and c.

What is the percent error $n$ in replacing the sine of $20^{\circ}$ by the value of the angle in radians? Repeat for the tangent of $20^{\circ}$; and explain the qualitative difference in the two error percentages.