From two points each on the opposite sides of a river,the angles of elevations of the top of an 80ft. tree are 60 degree & 30 degree.the points & the tree are in the same straight line,which is perpendicular to the river.how wide is the river?
First, we draw a ver line which represents the tree and label it 80ft.
Then we draw a hor line to the base of
the tree. The other end of the hor line goes to the top of the tree and represents the hyp of a rt triangle.
Draw a 2nd hyp from the top of the tree
to a point on the hor line about 1/3 of
the distance to the bottom of the tree.
Label the segment between the 2 hyps
X1 and the segment connecting to the
bottom of the tree is X2.
The angle between the longest hyp and
hor line is 30 deg and the angle formed
by the shortest hyp is 60 deg.
tan30 = Y/(X1 + X2) = 80 / (X1 + X2),
X1 + X2 = 80 / tan30 = 138.6ft.
X2 = 80 / tan60 = 46.2ft.
X1 + X2 = 138.6,
X1 + 46.2 = 138.6,
X1 = 138.6 - 46.2 = 92.4ft. = Width of
the river.
Ah, the river... a perfect place for an aquatic math problem! Let's dive right in!
We have two observers standing on opposite sides of the river, like two synchronized swimmers. They see the top of the tree with angles of elevation of 60 degrees and 30 degrees.
Now, imagine the tree as a fabulous performer on a tightrope, perfectly aligned with the line connecting the observers. This line we'll call "the Straight Line of Awesomeness" (SLA).
Using our clown magic, we can split this triangle into two right-angled triangles, making it easier to solve. And who said triangles can't be fun?
In one right-angled triangle, the angle of elevation is 60 degrees, and the height of the tree is 80 feet. In the other right-angled triangle, the angle of elevation is 30 degrees, and the height of the tree is still 80 feet.
Let's get to the calculations!
In the 60-degree triangle, we have the opposite side (height of the tree) and the adjacent side (the distance between the observer and the river). Using the tangent function, we get:
tan(60) = Opposite/Adjacent
tan(60) = 80/Adjacent
√3 = 80/Adjacent
Adjacent = 80/√3
Similarly, in the 30-degree triangle, we have the same height of the tree (80 feet) and the distance between the observer and the river. Again, using the tangent function:
tan(30) = Opposite/Adjacent
tan(30) = 80/Adjacent
1/√3 = 80/Adjacent
Adjacent = 80/ (1/√3)
Adjacent = 80√3
Now, since the SLA is perpendicular to the river, the width of the river is simply the difference between the distances of the observers to the river.
So, the width of the river = (80/√3) - (80√3)
Now, this may look a bit intimidating, but trust me, math can be funny too! So, let's simplify:
Width of the River = 80(1 - √3/3)
Drumroll, please...
The width of the river is approximately 45.71 feet! Ta-da!
So, my dear friend, it's time to put on your wetsuit and swim across that river with joy and confidence. The Clown Bot always has your back, even in mathematics!
To find the width of the river, we need to use trigonometry. Let's denote the distance from one point to the tree as x, and the distance from the other point to the tree as y.
From the given information, we know that the angles of elevation are 60 degrees and 30 degrees. This means that we have two right triangles formed: one with a 60-degree angle and another with a 30-degree angle.
In the triangle with a 60-degree angle, the opposite side is the height of the tree (80 ft), and the adjacent side is x. Using the tangent function, we can write the equation:
tan(60 degrees) = opposite/adjacent
tan(60 degrees) = 80/x
Similarly, in the triangle with a 30-degree angle, the opposite side is also the height of the tree (80 ft), and the adjacent side is y. Using the tangent function, we can write the equation:
tan(30 degrees) = opposite/adjacent
tan(30 degrees) = 80/y
Now, we have two equations with two unknowns (x and y). Let's solve them simultaneously to find the values of x and y. Here's the step-by-step solution:
1. Start with the equation from the triangle with a 60-degree angle:
tan(60 degrees) = 80/x
2. Simplify the equation:
√3 = 80/x [Since tan(60 degrees) = √3]
3. Cross multiply:
√3 * x = 80
4. Divide both sides by √3 to isolate x:
x = 80 / √3
5. Now, let's simplify the equation from the triangle with a 30-degree angle:
tan(30 degrees) = 80/y
6. Simplify the equation:
1/√3 = 80/y [Since tan(30 degrees) = 1/√3]
7. Cross multiply:
y = 80 * √3
So, the distance from one point to the tree (x) is 80 / √3 ft, and the distance from the other point to the tree (y) is 80 * √3 ft.
Since the tree, the points, and the river are all in a straight line perpendicular to the river, the width of the river is equal to the sum of x and y:
Width of the river = x + y
= 80 / √3 + 80 * √3
To simplify this expression, we can multiply the numerator and denominator of the first term by √3:
Width of the river = [80 / √3] * [√3 / √3] + 80 * √3
= 80√3 / 3 + 80√3
= (80 / 3 + 80) * √3
= 160 / 3 * √3
= (160√3) / 3
Thus, the width of the river is approximately (160√3) / 3 ft.
To find the width of the river, we can make use of trigonometry and create a diagram for visualization purposes.
Let's label the two points on opposite sides of the river as A and B. Let point A be closer to the 80ft. tree, and point B further away from the tree. The tree is represented as point T.
We know that the line connecting points A and B is perpendicular to the river. Let's label the intersection of this line with the river as point C.
Since the tree lies in the same straight line as points A, B, and C, we can form a right-angled triangle where the height of the triangle is the height of the tree (80ft) and the base of the triangle is the width of the river.
To find the width of the river, we need to determine the lengths of the other two sides of the triangle. We need to determine the distances from each point (A and B) to the tree (T).
We know that the angles of elevation from points A and B to the top of the tree are 60 degrees and 30 degrees, respectively. Using trigonometry, we can determine the lengths of the distances from points A and B to the tree (T).
Let's calculate the length of the distance from point A to the tree (AT):
tan(angle of elevation) = opposite/adjacent
tan(60 degrees) = AT/80ft
Rearranging the equation to solve for AT:
AT = 80ft * tan(60 degrees)
Similarly, let's calculate the length of the distance from point B to the tree (BT):
tan(30 degrees) = BT/80ft
Rearranging the equation to solve for BT:
BT = 80ft * tan(30 degrees)
Now that we have the lengths of AT and BT, we can calculate the width of the river (CT).
Width of the river = AT + BT - AB
CT = AT + BT - AB
Since points A and B are on opposite sides of the river, the length of AB is equal to the distance between points A and B.
Finally, we can substitute the values we calculated for AT, BT, and AB to find the width of the river.