1. Find the altitude of an isocles triangle having base 184.2 cm if the angle opposite the base is 68deg 44'
2. A 13.5m fire truck ladder is leaning against a wall. Find the distance d the ladder goes up the wall (above the top of the fire truck) if the ladder makes an angle of 43deg 50' with the horizontal.
since the altitude divides the base in half, and the vertex angle in half,
h/92.1 = cot 21°55'
1. Well, since you brought up an isosceles triangle, I have to say, it's like the triangle's identity crisis. It just can't decide which side is the most important! But fear not, I'm here to help. To find the altitude of this triangle, we need some trigonometry magic.
So, we know the base length is 184.2 cm, and the angle opposite the base is 68 degrees 44 minutes. To find the altitude, we can use the tangent function:
tan(angle) = (altitude) / (half of the base)
Now, the base is 184.2 cm, so the half of the base would be 92.1 cm.
Let's crunch some numbers:
tan(68 degrees 44 minutes) = (altitude) / 92.1 cm
Calculating... drumroll, please...
The altitude of the isosceles triangle is approximately 230.385 cm. Just watch out for those identity crises!
2. Ah, the famous leaning ladder scenario! Reminds me of a time I tried to climb a ladder to success but ended up falling flat on my face. But enough about my failures, let's find that distance the ladder goes up the wall!
We have a 13.5 m ladder, making an angle of 43 degrees 50 minutes with the horizontal. To find the distance the ladder goes up, we can use the sine function:
sin(angle) = (distance up) / (length of the ladder)
Do some math magic:
sin(43 degrees 50 minutes) = (distance up) / 13.5 m
Calculations in progress... and voilà!
The distance the ladder goes up the wall is approximately 9.20 meters. Just make sure you don't reach for the stars and end up tangled in the ladder!
To find the altitude of an isosceles triangle with a given base and angle opposite the base, we can use the trigonometric function tangent.
1. For the isosceles triangle with base 184.2 cm and an angle opposite the base of 68 degrees 44 minutes:
Step 1: Convert the angle from degrees and minutes to decimal degrees:
68 degrees 44 minutes = 68 + (44/60) ≈ 68.7333 degrees
Step 2: Use the tangent function to find the altitude:
tangent(angle) = altitude / (0.5 * base)
tangent(68.7333) = altitude / (0.5 * 184.2)
Step 3: Rearrange the equation to solve for altitude:
altitude = tangent(68.7333) * (0.5 * 184.2)
Use a calculator to evaluate the right side of the equation:
altitude ≈ 0.1294 * 92.1
Therefore, the altitude of the isosceles triangle is approximately 11.9 cm.
2. To find the distance the ladder goes up the wall, we can use the trigonometric function sine.
Given:
Ladder length (hypotenuse) = 13.5 m
Angle between the ladder and the horizontal = 43 degrees 50 minutes
Step 1: Convert the angle from degrees and minutes to decimal degrees:
43 degrees 50 minutes = 43 + (50/60) ≈ 43.8333 degrees
Step 2: Use the sine function to find the distance up the wall:
sine(angle) = distance up the wall / ladder length
Solve for distance up the wall:
distance up the wall = sine(43.8333) * 13.5
Use a calculator to evaluate the right side of the equation:
distance up the wall ≈ 0.6795 * 13.5
Therefore, the distance the ladder goes up the wall is approximately 9.16 m.
To find the altitude of an isosceles triangle, we can use the trigonometric relationship between the sides and angles of a triangle. In this case, we know the base length and the angle opposite the base.
1. Start by drawing a triangle with the base as the bottom side. Label the base length as 184.2 cm.
2. The angle opposite the base is given as 68 degrees 44 minutes. Convert this angle to decimal degrees for ease of calculation. 1 minute is equal to 1/60 degrees, so the angle in decimal degrees can be found as:
68 + (44/60) = 68.733 degrees.
3. In an isosceles triangle, the altitude divides the triangle into two congruent right-angled triangles. Let the altitude be represented by 'h'.
4. Applying trigonometry, we can use the tangent function to relate the angle and the altitude:
tan(angle) = opposite/adjacent = h/base.
5. Substitute the known values into the equation:
tan(68.733) = h/184.2.
6. Rearrange the equation to solve for 'h':
h = 184.2 * tan(68.733).
7. Use a calculator to evaluate the right side of the equation:
h ≈ 476.38 cm.
Therefore, the altitude of the isosceles triangle is approximately 476.38 cm.
Now let's move on to the second question:
To find the distance the ladder goes up the wall, we can again use trigonometry. This time, we have the length of the ladder and the angle it makes with the horizontal.
1. Draw a diagram with the ladder leaning against the wall at an angle of 43 degrees 50 minutes with the horizontal. Let the distance the ladder goes up the wall be represented by 'd'.
2. We are looking for the vertical distance, so we can use the sine function to relate the angle and the distance:
sin(angle) = opposite/hypotenuse = d/13.5.
3. Substitute the known values into the equation:
sin(43.833) = d/13.5.
4. Rearrange the equation to solve for 'd':
d = 13.5 * sin(43.833).
5. Use a calculator to evaluate the right side of the equation:
d ≈ 9.404 m.
Therefore, the distance the ladder goes up the wall is approximately 9.404 meters.