A circular ring has a cross-section as shown (left). If the outer radius is 22mm and the inner radius 20mm, calculate the cross-sectional area of the ring.

Two buildings on opposite sides of a street are $40 \mathrm{~m}$ apart. From the top of the taller building, which is $185 \mathrm{~m}$ high, the angle of depression to the top of the shorter building is $13^{\circ}$. Find the height of the shorter building.

Two observers are 400 ft apart on opposite sides of a tree. The angles of elevation from the observers to the top of the tree are 15° and 20° .Find the height of the tree.

The angle of elevation from a ship to the top of a lighthouse is

$$4 ^ { \circ }$$

If the top of the lighthouse is known to be 50 m above sea level, how far is the ship from the lighthouse?

A single slit has a width of

$$2.1 \times 10 ^ { - 6 } \mathrm { m }$$

and is used to form a diffraction pattern. Find the angle that locates the second dark fringe when the wavelength of the light is 660 nm.

A single slit has a width of

$$2.1 \times 10 ^ { - 6 } \mathrm { m }$$

and is used to form a diffraction pattern. Find the angle that locates the second dark fringe when the wavelength of the light is 430 nm.

Approximate the following integrals using Gaussian quadrature with $n=2,$ and compare your results to the exact values of the integrals. $a. \int_{1}^{1.5} x^{2} \ln x d x$ $b. \int_{0}^{1} x^{2} e^{-x} d x$ $c. \int_{0}^{0.35} \frac{2}{x^{2}-4} d x$ $d. \int_{0}^{\pi / 4} x^{2} \sin x d x$ $e. \int_{0}^{\pi / 4} e^{3 x} \sin 2 x d x$ $f. \int_{1}^{1.6} \frac{2 x}{x^{2}-4} d x$ $g. \int_{3}^{3.5} \frac{x}{\sqrt{x^{2}-4}} d x$ $h. \int_{0}^{\pi / 4}(\cos x)^{2} d x$

Use the sixth-degree Taylor polynomial (centered at 0) to approximate the definite integral. Function : $f(x)=\sqrt[3]{1+x^2}$
Definite Integral : $\int_0^{0.5} \sqrt[3]{1+x^2} d x$

Use the basic list of power series for common functions to find the power series for $f(x)=(1+x)^{2 / 3}$ centered at 0 .

Use the basic list of power series for common functions to find the power series for the function centered at c.
f(x) = ln x$^2$, c = 1

The volume of the material used to make the sphere and hemispherical bowl are the same. Given that the radius of the sphere is 7cm and the inner radius of the bowl is 10cm, calculate, to 1 d.p., the outer radius rcm of the bowl.

Calculate the number of sides a regular polygon has if an interior angle is five times the size of an exterior angle.

55% of students in a school are boys. a) What is the ratio of boys to girls? b) How many boys and how many girls are there if the school has 800 students?

$\frac{5}{9}$ of a litre carton of orange is pure orange juice, the rest is water. How many millilitres of each are in the carton?