1. A triangle has sides that are each 8 m long. Find the area of the triangle.
Solution:
2.
A cord of seasoned almond wood contains 128 cubic feet and
costs $190. How much should you pay for a pile of wood that is 5 feet
wide, 3 feet high and 8 feet long?For this problem, we know the lengths
of all of the sides of a triangle. The formula for finding
finding the area of a trianlge if we know the sides is 
where 
. For our
particular problem a, b, and c are all 8. Thus we have 
. So the
area of the triangle is 
or 27.7128 square meters.
where . For our
particular problem a, b, and c are all 8. Thus we have . So the
area of the triangle is or 27.7128 square meters.Solution:
3. In a right triangle having one angle of Our first step here is to find out the
volume of the pile of wood that we have. This stack of wood is in
the shape of a rectangular solid (block). Thus we find the volume
using the formula 
. Our pile of wood
has a volume of 
or 120 cubic feet. In order to determine what we should pay for
this pile of wood, we need to find out what fraction of a cord it
is. We do this by dividing our volume of wood by the volume of a
cord of wood. This will give us 
.
We now multiply this number by the price of a cord of wood to find out
how much we should pay for our pile of wood. This gives us 
. Thus we
should pay $178.13 for our pile of wood.
. Our pile of wood
has a volume of
or 120 cubic feet. In order to determine what we should pay for
this pile of wood, we need to find out what fraction of a cord it
is. We do this by dividing our volume of wood by the volume of a
cord of wood. This will give us .
We now multiply this number by the price of a cord of wood to find out
how much we should pay for our pile of wood. This gives us . Thus we
should pay $178.13 for our pile of wood.
, the length of the side
adjacent to the angle is 7 inches. What is the
length of the hypotenuse?Solution:
4. A circular swimming pool has a diameter of 45 feet and is
surrounded by a concrete sidewalk that is 5 feet wide. What is the area
of the sidewalk?This is a right triangle. We know
that the sides of a right triangle are related by trigonometric
functions. Our triangle here looks like 
.
We are being asked for the hypotenuse. In order to find this we
will use the cosine function of the angle given. This will give
us 
. When we
solve this equation, we get 
. The
hypotenuse if 14 inches.
.
We are being asked for the hypotenuse. In order to find this we
will use the cosine function of the angle given. This will give
us . When we
solve this equation, we get . The
hypotenuse if 14 inches.Solution:
5. A ladder is leaning against a building. If the bottom of the ladder
is For this problem, it might help to draw
a picture. This would look about like this 
.
We want to first find the area of the bigger circle. We then need
to find the area of the smaller circle. We will then subtract the
area of the smaller circle from the area of the bigger circle.
The radius of the smaller circle is 
. This will
make the radius of the larger circle 
. Thus the
area of the larger circle was 
. For the
smaller circle, the area is 
. Now we subtract the
area of the smaller circle from the area of the larger cirlce to get
the area of just the sidewalk 
. Thus the
area of the sidewalks is 785.398 square feet.
.
We want to first find the area of the bigger circle. We then need
to find the area of the smaller circle. We will then subtract the
area of the smaller circle from the area of the bigger circle.
The radius of the smaller circle is . This will
make the radius of the larger circle . Thus the
area of the larger circle was . For the
smaller circle, the area is . Now we subtract the
area of the smaller circle from the area of the larger cirlce to get
the area of just the sidewalk . Thus the
area of the sidewalks is 785.398 square feet.
feet from the wall and the top
of the ladder is 10 feet above the ground, how long is the ladder?Solution:
6. Ron Thiele bought an older house and wants to put in a new concrete
driveway. The driveway will be 36 feet long, 9 feet wide, and 6 inches
thick. Concrete is measured by the cubic yard. One sack of dry cement
mix costs $7.30, and it takes four sacks to mix up 1 cubic yard of
concrete. How much will it cost Ron to buy the cement?The picture for this problem will
look like a right triangle 
. We
can use the Pythagorean theorem to find the length of the ladder
(hypotenuse of the triangle). We will get 
. When
we simplify this we get 
.
Finally we have 
. The
ladder is 12.5 feet long.
. We
can use the Pythagorean theorem to find the length of the ladder
(hypotenuse of the triangle). We will get . When
we simplify this we get .
Finally we have . The
ladder is 12.5 feet long.Solution:7. A 5.4-foot-tall woman casts a shadow of 2 feet at the same instant that a telephone pole casts a shadow of 9 feet. How tall is the pole?
For this problem we need to find the volume of concrete that will be needed for the driveway. Since concrete is measured by the cubic yard, it will be easiest to change all the driveway measurements into yards before calculating. The length of the driveway in yards is. The width of the driveway in yards is
. The thickness of the driveway in yards is
. The volume (number of cubic yards) of the driveway is
. We need to use 4 bags of concrete for each cubic yard. Thus we need 24 bags (
) of concrete. Since each bag of concrete costs $7.30, we need to multiply the number of bags needed by the price per bag to find out how must the driveway will cost. Thus the cost is 7.30 * 24 = $175.20
Solution:8. The Sears Tower (in Chicago) is the tallest building in the United States. From a point 900 feet from the building, you measure the angle of elevation of the top of the building. If the angle is
This problem requires the use of similar triangles. The two triangle will look like
and
.
We now need to form the ratios to be able to find the length of the pole. These ratios will be. When we solve this, we get
. Thus the telephone pole is 24.3 feet tall.
, how tall (to the nearest foot)
is the Sears Tower?Solution:9. You jog
For this problem, we will again make use of a right triangle and use a trigonometric function. The picture for this problem looks like
The height of the building can be found using the ratio of the building divided by the distance from the building which we know is equal to the tangent of the angle of elevation. Thus we have
. The Sears Tower is 1453.82 feet.
mile due north, then jog 
miles due east, and then return
to your starting point via a straight line path. How many miles have
your jogged?Solution:
For this problem a picture will be helpful.
We need to find the lengths of all of the sides of the triangle that was jogged. We can find the last side using the Pythagorean theorem. This will look like. The last side is equal to 1.677 miles. We now add together all the sides of the triangle jogged to get .75 + 1.5 + 1.677 = 3.927 total miles jogged.
