Chapter 2

Find the perimeter of each figure. Square: side of \(85 \mathrm{m}\)

Find the perimeter of each figure. Rhombus: side of \(2.46 \mathrm{ft}\)

Find the perimeter of each figure. Rectangle: \(l=9.200\) in. \(, w=7.420\) in.

Find the perimeter of each figure. $$\text { Rectangle: } l=142 \mathrm{cm}, w=126 \mathrm{cm}$$

Find the circumference of the circle with the given radius or diameter. $$r=275 \mathrm{ft}$$

Find the circumference of the circle with the given radius or diameter. $$r=0.563 \mathrm{m}$$

Calculate the indicated areas. All data are accurate to at least two significant digits. Using aerial photography, the widths of an area burned by a forest fire were measured at \(0.5-\mathrm{mi}\) intervals, as shown in the following table: $$\begin{array}{l|c|c|c|c|c|c|c|c|c}\text {Distance (mi)} & 0.0 & 0.5 & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & 4.0 \\\\\hline \text {Width (mi)} & 0.6 & 2.2 & 4.7 & 3.1 & 3.6 & 1.6 & 2.2 & 1.5 & 0.8\end{array}$$ Determine the area burned by the fire by using the trapezoidal rule.

Find the circumference of the circle with the given radius or diameter. $$d=23.1 \mathrm{mm}$$

Find the circumference of the circle with the given radius or diameter. \(d=8.2\) in.

Find the area of the circle with the given radius or diameter. $$r=0.0952 \mathrm{yd}$$

Find the area of each triangle. Right triangle with legs \(3.46 \mathrm{ft}\) and \(2.55 \mathrm{ft}\)

Find the area of each figure. Square: \(s=6.4 \mathrm{mm}\)

Calculate the indicated areas. All data are accurate to at least two significant digits. The widths (in \(\mathrm{m}\) ) of half the central arena in the Colosseum in Rome are shown in the following table, starting at one end and measuring from the middle to one side at \(4.0-\mathrm{m}\) intervals. Find the area of the arena by the trapezoidal rule. Hint: Remember to double the distances. $$\begin{array}{l|r|r|r|l|l|l}\text {Dist.from middle (m)} & 0.0 & 4.0 & 8.0 & 12.0 & 16.0 & 20.0 \\\\\hline \text {Width (m)} & 55.0 & 54.8 & 54.0 & 53.6 & 51.2 & 49.0\end{array}$$ $$\begin{array}{l|l|l|l|l|l|l}\text {Dist.} & 24.0 & 28.0 & 32.0 & 36.0 & 40.0 & 44.0 \\\\\hline \text {Width} & 45.8 & 42.0 & 37.2 & 31.1 & 21.7 & 0.0\end{array}$$

Find the volume or area of each solid figure for the given values. See Figs. 2.112 to 2.119 . Volume of right prism: square base of side \(29.0 \mathrm{cm}, h=11.2 \mathrm{cm}\)

Find the area of the circle with the given radius or diameter. $$r=45.8 \mathrm{cm}$$

Find the area of each figure. $$\text { Square: } s=15.6 \mathrm{ft}$$

Find the area of each triangle. Right triangle with legs \(234 \mathrm{mm}\) and \(342 \mathrm{mm}\)

Find the area of the circle with the given radius or diameter. $$d=2.33 \mathrm{m}$$

Find the area of each figure. Rectangle: \(l=8.35\) in. \(, w=2.81\) in.

Find the area of each triangle. Isosceles triangle, equal sides of \(0.986 \mathrm{m}\), third side of \(0.884 \mathrm{m}\)

Find the area of the circle with the given radius or diameter. $$d=1256 \mathrm{ft}$$

Find the area of each figure. $$\text { Rectangle: } l=142 \mathrm{cm}, w=126 \mathrm{cm}$$

Find the area of each triangle. Equilateral triangle of sides 3200 yd

Calculate the indicated areas. All data are accurate to at least two significant digits. Soundings taken across a river channel give the following depths with the corresponding distances from one shore. $$\begin{array}{l|l|l|l|l|l|l|l|l|l|l|l}\text {Distance (ft)} & 0 & 50 & 100 & 150 & 200 & 250 & 300 & 350 & 400 & 450 & 500 \\\\\hline \text {Depth (ft)} & 5 & 12 & 17 & 21 & 22 & 25 & 26 & 16 & 10 & 8 & 0\end{array}$$ Find the area of the cross section of the channel using Simpson's rule.

Find the area of the circle with the given circumference. $$c=40.1 \mathrm{cm}$$

Find the area of the circle with the given circumference. $$c=147 \mathrm{m}$$

Calculate the area of the circle by the indicated method. The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter. $$\begin{array}{l|l|l|l|l|l|l|l}\text {Distance (in.)} & 0.000 & 0.250 & 0.500 & 0.750 & 1.000 & 1.250 & 1.500 & 1.750 & 2.000 \\\\\hline \text {Length (in.)} & 0.000 & 1.323 & 1.732 & 1.936 & 2.000 & 1.936 & 1.732 & 1.323 & 0.000\end{array}$$ Using the formula \(A=\pi r^{2},\) the area of the circle is 3.14 in. \(^{2}\). Find the area of the circle using the trapezoidal rule and only the values of distance of 0.000 in. 0.500 in., 1.000 in., 1.500 in., and 2.000 in. with the corresponding values of the chord lengths. Explain why the value found is less than 3.14 in. \(^{2}\).

Find the perimeter of each triangle. An equilateral triangle of sides \(21.5 \mathrm{cm}\)

Find the perimeter of each triangle. Isosceles triangle, equal sides of 2.45 in., third side of 3.22 in.

Calculate the area of the circle by the indicated method. The lengths of parallel chords of a circle that are 0.250 in. apart are given in the following table. The diameter of the circle is 2.000 in. The distance shown is the distance from one end of a diameter. $$\begin{array}{l|l|l|l|l|l|l|l}\text {Distance (in.)} & 0.000 & 0.250 & 0.500 & 0.750 & 1.000 & 1.250 & 1.500 & 1.750 & 2.000 \\\\\hline \text {Length (in.)} & 0.000 & 1.323 & 1.732 & 1.936 & 2.000 & 1.936 & 1.732 & 1.323 & 0.000\end{array}$$ Using the formula \(A=\pi r^{2},\) the area of the circle is 3.14 in. \(^{2}\). Find the area of the circle using Simpson's rule and all values in the table. Explain why the value found is closer to 3.14 in. \(^{2}\) than the value found in Exercise 21.

\(\text {Solve the given problems.}\) Derive a formula for the total surface area \(A\) of a hemispherical volume of radius \(r\) (curved surface and flat surface).

\(\text {Solve the given problems.}\) The radius of a cylinder is twice as long as the radius of a cone, and the height of the cylinder is half as long as the height of the cone. What is the ratio of the volume of the cylinder to that of the cone?

Solve the given problems. If the angle between adjacent sides of a parallelogram is \(90^{\circ},\) what conclusion can you make about the parallelogram?

\(\text {Solve the given problems.}\) The base area of a cone is one-fourth of the total area. Find the ratio of the radius to the slant height.

\(\text {Solve the given problems.}\) In designing a spherical weather balloon, it is decided to double the diameter of the balloon so that it can carry a heavier instrument load. What is the ratio of the final surface area to the original surface area?

Change the given angles to radian measure. $$22.5^{\circ}$$

Solve the given problems. Find the area of a square whose diagonal is \(24.0 \mathrm{cm}\).

\(\text {Solve the given problems.}\) During a rainfall of 1.00 in., what weight of water falls on an area of \(1.00 \mathrm{mi}^{2} ?\) Each cubic foot of water weighs \(62.4 \mathrm{lb}\).

Change the given angles to radian measure. $$60.0^{\circ}$$

\(\text {Solve the given problems.}\) A rectangular box is to be used to store radioactive materials. The inside of the box is 12.0 in. long. 9.50 in. wide, and 8.75 in. deep. What is the area of sheet lead that must be used to line the inside of the box?

Change the given angles to radian measure. $$125.2^{\circ}$$

\(\text {Solve the given problems.}\) A swimming pool is \(50.0 \mathrm{ft}\) wide, \(78.0 \mathrm{ft}\) long. \(3.50 \mathrm{ft}\) deep at one end, and \(8.75 \mathrm{ft}\) deep at the other end. How many cubic feet of water can it hold? (The slope on the bottom is constant.) See Fig. 2.122

Change the given angles to radian measure. $$323.0^{\circ}$$

Solve the given problems. The sum \(S\) of the measures of the interior angles of a polygon with \(n\) sides is \(S=180(n-2) .\) (a) Solve for \(n .\) (b) If \(S=3600^{\circ}\), how many sides does the polygon have?

\(\text {Solve the given problems.}\) The Alaskan oil pipeline is \(750 \mathrm{mi}\) long and has a diameter of \(4.0 \mathrm{ft}\) What is the maximum volume of the pipeline?

Solve the given problems. What is the angle between the bisectors of the acute angles of a right triangle?

\(\text {Solve the given problems.}\) The volume of a frustum of a pyramid is \(V=\frac{1}{3} h\left(a^{2}+a b+b^{2}\right)\) (see Fig. 2.123 ). (This equation was discovered by the ancient Egyptians.) If the base of a statue is the frustum of a pyramid, find its volume if \(\quad a=2.50 \mathrm{m}, b=3.25 \mathrm{m}, \quad\) and \(h=0.750 \mathrm{m}\)

Solve the given problems. If the midpoints of the sides of an isosceles triangle are joined, another triangle is formed. What do you conclude about this inner triangle?

\(\text {Solve the given problems.}\) A pole supporting a wind turbine is constructed of solid steel and is in the shape of a frustum of a cone. It measures \(62.5 \mathrm{m}\) high, and the diameter of the pole at the bottom and top are \(3.88 \mathrm{m}\) and 1.90 \(\mathrm{m}\), respectively. What is the volume of the pole?

Solve the given problems. For what type of triangle is the centroid the same as the intersection of altitudes and the intersection of angle bisectors?