Math / Geometry

QuestionTo find the height $h$ of Mount St. Melon in the Cantaloupe Mountains, two angle measurements were taken 1200 feet apart along a direct line toward the mountain. Using these measurements, find the height of the mountain. Homework Help

Studdy Solution

STEP 1

What is this asking? We need to figure out how tall Mount St.
Melon is using two angles measured from different spots a known distance apart. Watch out! Don't mix up the angles and distances!
Make sure you're using the right measurements for the right parts of the triangles.

STEP 2

1. Set up the problem
2. Find the distance to the mountain
3. Calculate the height

STEP 3

Alright, imagine we're looking at this mountain from two different spots.
We've got two angles, $35^\circ$ and $42^\circ$ , and we know the distance between those spots is 1200 feet.
Let's call the distance from the closer spot to the base of the mountain $x$ .

STEP 4

We can actually think of this as two right triangles!
One big one with the $35^\circ$ angle, and a smaller one inside it with the $42^\circ$ angle.
Both triangles share the same height, which is what we want to find.

STEP 5

Let's use some trigonometry!
For the bigger triangle with the $35^\circ$ angle, we know that $\tan(35^\circ) = \frac{h}{x + 1200}$ .
Remember, tangent is opposite over adjacent.

STEP 6

For the smaller triangle with the $42^\circ$ angle, we have $\tan(42^\circ) = \frac{h}{x}$ .
Same idea, tangent is opposite over adjacent.

STEP 7

Now we have two equations with two unknowns, $h$ and $x$ .
Let's solve for $x$ first.
From the second equation, we get $h = x \cdot \tan(42^\circ)$ .
Let's substitute this into the first equation: $\tan(35^\circ) = \frac{x \cdot \tan(42^\circ)}{x + 1200}$ .

STEP 8

Time to rearrange!
Multiply both sides by $x + 1200$ to get $\tan(35^\circ)(x + 1200) = x \cdot \tan(42^\circ)$ .
Distribute the $\tan(35^\circ)$ to get $x \cdot \tan(35^\circ) + 1200 \cdot \tan(35^\circ) = x \cdot \tan(42^\circ)$ .

STEP 9

Now, move the $x$ terms to one side: $1200 \cdot \tan(35^\circ) = x \cdot \tan(42^\circ) - x \cdot \tan(35^\circ)$ .
Factor out the $x$ to get $1200 \cdot \tan(35^\circ) = x(\tan(42^\circ) - \tan(35^\circ))$ .

STEP 10

Finally, divide both sides by $(\tan(42^\circ) - \tan(35^\circ))$ to isolate $x$ : $x = \frac{1200 \cdot \tan(35^\circ)}{\tan(42^\circ) - \tan(35^\circ)}$ .
Plugging in the values for the tangents, we get $x = \frac{1200 \cdot 0.7002}{0.9004 - 0.7002} = \frac{840.24}{0.2002} \approx 4196.9$ .
So, $x \approx \textbf{4196.9 feet}$ .

STEP 11

Now that we know $x$ , we can plug it back into either of our original equations to find $h$ .
Let's use the simpler one: $h = x \cdot \tan(42^\circ)$ .

STEP 12

Substituting our value for $x$ , we get $h = 4196.9 \cdot \tan(42^\circ) = 4196.9 \cdot 0.9004 \approx 3778.9$ .

STEP 13

The height of Mount St.
Melon is approximately 3778.9 feet!

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QuestionTo find the height $h$ of Mount St. Melon in the Cantaloupe Mountains, two angle measurements were taken 1200 feet apart along a direct line toward the mountain. Using these measurements, find the height of the mountain. Homework Help

Studdy Solution

STEP 1

What is this asking? We need to figure out how tall Mount St.
Melon is using two angles measured from different spots a known distance apart. Watch out! Don't mix up the angles and distances!
Make sure you're using the right measurements for the right parts of the triangles.

STEP 2

1. Set up the problem
2. Find the distance to the mountain
3. Calculate the height

STEP 3

Alright, imagine we're looking at this mountain from two different spots.
We've got two angles, $35^\circ$ and $42^\circ$ , and we know the distance between those spots is 1200 feet.
Let's call the distance from the closer spot to the base of the mountain $x$ .

STEP 4

We can actually think of this as two right triangles!
One big one with the $35^\circ$ angle, and a smaller one inside it with the $42^\circ$ angle.
Both triangles share the same height, which is what we want to find.

STEP 5

Let's use some trigonometry!
For the bigger triangle with the $35^\circ$ angle, we know that $\tan(35^\circ) = \frac{h}{x + 1200}$ .
Remember, tangent is opposite over adjacent.

STEP 6

For the smaller triangle with the $42^\circ$ angle, we have $\tan(42^\circ) = \frac{h}{x}$ .
Same idea, tangent is opposite over adjacent.

STEP 7

Now we have two equations with two unknowns, $h$ and $x$ .
Let's solve for $x$ first.
From the second equation, we get $h = x \cdot \tan(42^\circ)$ .
Let's substitute this into the first equation: $\tan(35^\circ) = \frac{x \cdot \tan(42^\circ)}{x + 1200}$ .

STEP 8

Time to rearrange!
Multiply both sides by $x + 1200$ to get $\tan(35^\circ)(x + 1200) = x \cdot \tan(42^\circ)$ .
Distribute the $\tan(35^\circ)$ to get $x \cdot \tan(35^\circ) + 1200 \cdot \tan(35^\circ) = x \cdot \tan(42^\circ)$ .

STEP 9

Now, move the $x$ terms to one side: $1200 \cdot \tan(35^\circ) = x \cdot \tan(42^\circ) - x \cdot \tan(35^\circ)$ .
Factor out the $x$ to get $1200 \cdot \tan(35^\circ) = x(\tan(42^\circ) - \tan(35^\circ))$ .

STEP 10

STEP 11

Now that we know $x$ , we can plug it back into either of our original equations to find $h$ .
Let's use the simpler one: $h = x \cdot \tan(42^\circ)$ .

STEP 12

Substituting our value for $x$ , we get $h = 4196.9 \cdot \tan(42^\circ) = 4196.9 \cdot 0.9004 \approx 3778.9$ .

STEP 13

The height of Mount St.
Melon is approximately 3778.9 feet!

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QuestionTo find the height hhh of Mount St. Melon in the Cantaloupe Mountains, two angle measurements were taken 1200 feet apart along a direct line toward the mountain. Using these measurements, find the height of the mountain. Homework Help

Studdy Solution

STEP 1

What is this asking? We need to figure out how tall Mount St.Melon is using two angles measured from different spots a known distance apart. Watch out! Don't mix up the angles and distances!Make sure you're using the right measurements for the right parts of the triangles.

STEP 2

1. Set up the problem2. Find the distance to the mountain3. Calculate the height

STEP 3

Alright, imagine we're looking at this mountain from two different spots.We've got two angles, 35∘35^\circ35∘ and 42∘42^\circ42∘, and we know the distance between those spots is **1200 feet**.Let's call the distance from the closer spot to the base of the mountain xxx.

STEP 4

We can actually think of this as two right triangles!One big one with the 35∘35^\circ35∘ angle, and a smaller one inside it with the 42∘42^\circ42∘ angle.Both triangles share the same height, which is what we want to find.

STEP 5

Let's use some trigonometry!For the bigger triangle with the 35∘35^\circ35∘ angle, we know that tan⁡(35∘)=hx+1200\tan(35^\circ) = \frac{h}{x + 1200}tan(35∘)=x+1200h​.Remember, tangent is opposite over adjacent.

STEP 6

For the smaller triangle with the 42∘42^\circ42∘ angle, we have tan⁡(42∘)=hx\tan(42^\circ) = \frac{h}{x}tan(42∘)=xh​.Same idea, tangent is opposite over adjacent.

STEP 7

STEP 8

STEP 9

STEP 10

STEP 11

Now that we know xxx, we can plug it back into either of our original equations to find hhh.Let's use the simpler one: h=x⋅tan⁡(42∘)h = x \cdot \tan(42^\circ)h=x⋅tan(42∘).

STEP 12

Substituting our value for xxx, we get h=4196.9⋅tan⁡(42∘)=4196.9⋅0.9004≈3778.9h = 4196.9 \cdot \tan(42^\circ) = 4196.9 \cdot 0.9004 \approx 3778.9h=4196.9⋅tan(42∘)=4196.9⋅0.9004≈3778.9.

STEP 13

The height of Mount St.Melon is approximately **3778.9 feet**!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionTo find the height $h$ of Mount St. Melon in the Cantaloupe Mountains, two angle measurements were taken 1200 feet apart along a direct line toward the mountain. Using these measurements, find the height of the mountain. Homework Help

What is this asking? We need to figure out how tall Mount St.
Melon is using two angles measured from different spots a known distance apart. Watch out! Don't mix up the angles and distances!
Make sure you're using the right measurements for the right parts of the triangles.

1. Set up the problem
2. Find the distance to the mountain
3. Calculate the height

Alright, imagine we're looking at this mountain from two different spots.
We've got two angles, $35^\circ$ and $42^\circ$ , and we know the distance between those spots is 1200 feet.
Let's call the distance from the closer spot to the base of the mountain $x$ .

We can actually think of this as two right triangles!
One big one with the $35^\circ$ angle, and a smaller one inside it with the $42^\circ$ angle.
Both triangles share the same height, which is what we want to find.

Let's use some trigonometry!
For the bigger triangle with the $35^\circ$ angle, we know that $\tan(35^\circ) = \frac{h}{x + 1200}$ .
Remember, tangent is opposite over adjacent.

For the smaller triangle with the $42^\circ$ angle, we have $\tan(42^\circ) = \frac{h}{x}$ .
Same idea, tangent is opposite over adjacent.

Now that we know $x$ , we can plug it back into either of our original equations to find $h$ .
Let's use the simpler one: $h = x \cdot \tan(42^\circ)$ .

Substituting our value for $x$ , we get $h = 4196.9 \cdot \tan(42^\circ) = 4196.9 \cdot 0.9004 \approx 3778.9$ .

The height of Mount St.
Melon is approximately 3778.9 feet!