(a) decide whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

Recent sales of some real estate and record profits make it possible for a manufacturer to set aside $\$ 800,000$ in a fund to be used for modernization and remodeling. How much can be withdrawn from this fund at the beginning of each half-year for the next $3$ years if the fund earns $7.7 \%$ compounded semiannually?

Express the quadratic form in the matrix notation

$$x^TAx$$

, where A is a symmetric matrix.

$$(a) 5x_1^2+5x_1x_2 (b) -7x_1x_2 (c) x_1^2+x_2^2-3x_3^2-5x_1x_2+9x_1x_3$$

Express the quadratic equation in the matrix form

$$x^TAx + Kx + f = 0$$

, where

$$x^TAx$$

is the associated quadratic form and K is an appropriate matrix. (a) 2x² + xy + x - 6y + 2 = 0 (b) y² + 7x - 8y - 5 = 0

(a) state whether the problem relates to an ordinary annuity or an annuity due and then (b) solve the problem.

A couple has determined that they need $\$ 1.5$ million to establish an annuity when they retire in $25$ years. How much money should they deposit at the end of each month in an investment plan that pays $10 \%$ compounded monthly, so that they will have the $\$ 1.5 \mathrm{mil}$ lion in $25$ years?

The pilot of an airplane wishes to fly due north, but there is a 36 km/h wind blowing toward the west. (a) In what direction should the pilot head her plane if its speed relative to the air is 350 km/h? (b) Draw a vector diagram that illustrates your result in part (a). (c) If the pilot decreases the airspeed of the plane, but still wants to head due north, should she increase or decrease the angle found in part (a)?

A jet is flying through a wind that is blowing with a speed of 55mi/h in the direction $\mathrm{N} 30^{\circ} \mathrm{E}$ (see the figure). The jet has a speed of $765 \mathrm{mi} / \mathrm{h}$ relative to the air, and the pilot heads the jet in the direction $\mathrm{N} 45^{\circ} \mathrm{E}$.

(a) Express the velocity of the wind as a vector in component form.

(b) Express the velocity of the jet relative to the air as a vector in component form.

(c) Find the true velocity of the jet.

(d) Find the true speed and direction of the jet.

A pilot heads his jet due east. The jet has a speed of 425 mi/h relative to the air. The wind is blowing due north with a speed of 40 mi/h. (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet.

A jet is flying through a wind that is blowing with a speed of 55 mi/h in the direction N 30 degrees E . The jet has a speed of 765 mi/h relative to the air, and the pilot heads the jet in the direction N 45 degrees E.

Find the true speed and direction of the jet.

A debt of $\$ 8000$ is to be amortized with $8$ equal semiannual payments. If the interest rate is $12 \%$ compounded semiannually, what is the size of each payment?

Write the first four terms and the 10 th term of the sequence whose $n$th term is

$$a_n=\frac{n-4}{n(n+2)}$$

Find the $80$th term of the arithmetic sequence with first term $-2$ and common difference $3$.

The number N of trees of a given species per acre is approximated by the model N = 68(10^-0.04X), 5 ≤ x ≤ 40, where x is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when N = 21.

A baseball player has a lifetime batting average of $0.300$. If each at bat is independent and he comes to bat $5$ times in a given game, what is the probability that

(a) he will get $3$ hits?

(b) he will get more than $3$ hits?

An open box is made from a square piece of cardboard 30 inches on a side by cutting identical squares from the corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of the square cut from each corner, x.