use the pythagorean theorem
write an equation that can be used to answer the question. then
solve. round to the nearest tenth if necessary.
1. how far is the ship from the lighthouse?
2. how long is the wire supporting the sign?
3. how far above the water is the person parasailing?
4. how wide is the pond?
5. how high is the ramp?
6. how high is the end of the ladder against the building?
7. geography suppose birmingham, huntsville, and gadsden, alabama, form a right triangle. what is the distance from huntsville to gadsden? round to the nearest tenth if necessary.
8. geometry find the diameter ( d ) of the circle in the figure at the right. round to the nearest tenth if necessary.

Question

Accepted Answer

Let's solve problem 1 first (How far is the ship from the lighthouse?):

# Explanation:
## Step1: Identify the right triangle sides
We have a right triangle where one leg (horizontal) is 8 m, another leg (vertical) is 6 m, and the hypotenuse $ c $ is the distance from the ship to the lighthouse. By the Pythagorean theorem, $ a^2 + b^2 = c^2 $, where $ a = 6 $, $ b = 8 $.
## Step2: Substitute values and calculate
Substitute $ a = 6 $, $ b = 8 $ into the formula: $ 6^2 + 8^2 = c^2 $ → $ 36 + 64 = c^2 $ → $ 100 = c^2 $. Take the square root: $ c = \sqrt{100} = 10 $.
# Answer:
The ship is 10 meters from the lighthouse.

Now problem 2 (How long is the wire supporting the sign?):

# Explanation:
## Step1: Identify the right triangle sides
The vertical segment from the wire's attachment to the sign's top is 1.5 ft, the horizontal segment from the sign to the pole is 2 ft. The wire is the hypotenuse $ c $. Using Pythagorean theorem: $ a^2 + b^2 = c^2 $, $ a = 1.5 $, $ b = 2 $.
## Step2: Substitute and compute
$ 1.5^2 + 2^2 = c^2 $ → $ 2.25 + 4 = c^2 $ → $ 6.25 = c^2 $. Square root: $ c = \sqrt{6.25} = 2.5 $.
# Answer:
The wire is 2.5 feet long.

Problem 3 (How far above the water is the person parasailing?):

# Explanation:
## Step1: Identify the right triangle sides
The hypotenuse of the right triangle is 100 yd, one leg (horizontal) is 80 yd. Let the vertical leg (height above water) be $ a $. By Pythagorean theorem: $ a^2 + 80^2 = 100^2 $.
## Step2: Solve for $ a $
$ a^2 = 100^2 - 80^2 $ → $ a^2 = 10000 - 6400 = 3600 $. Take square root: $ a = \sqrt{3600} = 60 $.
# Answer:
The person is 60 yards above the water.

Problem 4 (How wide is the pond?):

# Explanation:
## Step1: Identify the right triangle sides
The hypotenuse is 120 ft, one leg (horizontal) is 95 ft. Let the vertical leg (width of pond) be $ w $. Using Pythagorean theorem: $ 95^2 + w^2 = 120^2 $.
## Step2: Solve for $ w $
$ w^2 = 120^2 - 95^2 $ → $ w^2 = 14400 - 9025 = 5375 $. Take square root: $ w \approx \sqrt{5375} \approx 73.3 $ (rounded to nearest tenth).
# Answer:
The pond is approximately 73.3 feet wide.

Problem 5 (How high is the ramp?):

# Explanation:
## Step1: Identify the right triangle sides
The hypotenuse of the ramp triangle is 21 ft, one leg (horizontal) is 19 ft. Let the vertical leg (height of ramp) be $ h $. By Pythagorean theorem: $ h^2 + 19^2 = 21^2 $.
## Step2: Solve for $ h $
$ h^2 = 21^2 - 19^2 $ → $ h^2 = 441 - 361 = 80 $. Take square root: $ h \approx \sqrt{80} \approx 8.9 $ (rounded to nearest tenth).
# Answer:
The ramp is approximately 8.9 feet high.

Problem 6 (How high is the end of the ladder against the building?):

# Explanation:
## Step1: Identify the right triangle sides
The ladder (hypotenuse) is 13 ft, the base (horizontal) is 4 ft. Let the height (vertical) be $ h $. Using Pythagorean theorem: $ h^2 + 4^2 = 13^2 $.
## Step2: Solve for $ h $
$ h^2 = 13^2 - 4^2 $ → $ h^2 = 169 - 16 = 153 $. Take square root: $ h \approx \sqrt{153} \approx 12.4 $ (rounded to nearest tenth).
# Answer:
The ladder is approximately 12.4 feet high against the building.

Problem 7 (Distance from Huntsville to Gadsden):

# Explanation:
## Step1: Identify the right triangle sides
The right triangle has legs 98 mi (Birmingham to Huntsville) and 61 mi (Birmingham to Gadsden). The hypotenuse? Wait, no—wait, the right angle is at Gadsden, so Huntsville to Gadsden is a leg? Wait, no: Birmingham, Huntsville, Gadsden form a right triangle with right angle at Gadsden. So Birmingham to Huntsville is one leg (98 mi), Birmingham to Gadsden is another leg (61 mi), and Huntsville to Gadsden is the hypotenuse? Wait, no—wait, the diagram: Huntsville, Gadsden, Birmingham. Right angle at Gadsden. So sides: Birmingham to Gadsden: 61 mi, Birmingham to Huntsville: 98 mi, Huntsville to Gadsden: $ d $ (hypotenuse? No, wait, right angle at Gadsden, so Huntsville to Gadsden and Birmingham to Gadsden are legs, Birmingham to Huntsville is hypotenuse? Wait, no, the problem says "distance from Huntsville to Gadsden". Wait, let's recheck: the triangle is right-angled at Gadsden. So sides: Huntsville to Gadsden: $ d $, Gadsden to Birmingham: 61 mi, Huntsville to Birmingham: 98 mi? No, wait the diagram: Huntsville at top, Gadsden at right (right angle), Birmingham at bottom. So Huntsville to Birmingham: 98 mi (vertical leg), Birmingham to Gadsden: 61 mi (horizontal leg), Huntsville to Gadsden: $ d $ (hypotenuse). Wait, no—Pythagorean theorem: $ d^2 + 61^2 = 98^2 $? No, wait, if right angle at Gadsden, then Huntsville-Gadsden and Gadsden-Birmingham are legs, Huntsville-Birmingham is hypotenuse. Wait, the problem is "distance from Huntsville to Gadsden", so that's a leg? Wait, no, maybe I got the legs reversed. Wait, let's read again: "Birmingham, Huntsville, and Gadsden, Alabama, form a right triangle. What is the distance from Huntsville to Gadsden?". The diagram shows Huntsville at top, Gadsden at right (right angle), Birmingham at bottom. So Birmingham to Gadsden: 61 mi (horizontal), Birmingham to Huntsville: 98 mi (vertical), so Huntsville to Gadsden is the hypotenuse? Wait, no, 98 and 61 are legs, hypotenuse is Huntsville to Gadsden? Wait, no, 98^2 + 61^2 = d^2? Wait, no, if right angle at Gadsden, then Huntsville to Gadsden (d) and Gadsden to Birmingham (61) are legs, Huntsville to Birmingham (98) is hypotenuse. So: $ d^2 + 61^2 = 98^2 $ → $ d^2 = 98^2 - 61^2 $. Let's compute: 98²=9604, 61²=3721, so 9604 - 3721 = 5883. Then $ d = \sqrt{5883} \approx 76.7 $ (rounded to nearest tenth).
## Step1: Apply Pythagorean theorem
Right triangle with legs $ d $ (Huntsville to Gadsden) and 61 mi (Birmingham to Gadsden), hypotenuse 98 mi (Birmingham to Huntsville). So $ d^2 + 61^2 = 98^2 $.
## Step2: Solve for $ d $
$ d^2 = 98^2 - 61^2 = (98 - 61)(98 + 61) = 37 \times 159 = 5883 $. $ d = \sqrt{5883} \approx 76.7 $ (rounded to nearest tenth).
# Answer:
The distance from Huntsville to Gadsden is approximately 76.7 miles.

Problem 8 (Find the diameter $ d $ of the circle):

# Explanation:
## Step1: Identify the right triangle
The triangle is right-angled (since the radius is perpendicular to the tangent, so the triangle with legs 18 ft (horizontal), $ d/2 $ (radius, vertical), and hypotenuse 22 ft. Wait, the diameter is $ d $, so radius is $ d/2 $. The right triangle has legs 18 ft and $ d/2 $, hypotenuse 22 ft. So by Pythagorean theorem: $ (d/2)^2 + 18^2 = 22^2 $.
## Step2: Solve for $ d $
First, $ (d/2)^2 = 22^2 - 18^2 $ → $ (d/2)^2 = 484 - 324 = 160 $. Then $ d/2 = \sqrt{160} \approx 12.649 $. Then $ d \approx 2 \times 12.649 \approx 25.3 $ (rounded to nearest tenth).
# Answer:
The diameter $ d $ is approximately 25.3 feet.

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Identify the right triangle sides

Step2: Substitute values and calculate

Step1: Identify the right triangle sides

Step2: Substitute and compute

Step1: Identify the right triangle sides

Step2: Solve for \( a \)

Answer: