while visiting a mountain, John approximated the angle of elevation to the top of a hill to be 25 degrees. Alfter walking 350 fe
et closer he guessed that the angle of elevation had increased by 14 degrees. Approx. how tall is the hill?
1 answer:
There are several ways of going about this problem, but just know that it all boils down to triangles and the rules/laws of triangles.
So we start by making a large right triangle, because the hill is vertical with the horizontal ground, with 25° as the left angle (where John looks up to top), 90° is the right angle (where ground meets hill base). So we see that the top side of the triangle is the hypotenuse, and equals the line of sight from John to hilltop.
Now we've got additional information that if John walks 350ft towards the hill, his angle of elevation increases by 14. So that = 25+14 = 39. How does that possibly help us?? Well now we can make 2 triangles inside of the one we've already made. So that now we have the triangle base split between the left angle of 25° and where he stopped 350ft to the right of that.
Now the supplement of 39 is 141, and the remaining piece of that too left triangle = 180-25-141 = 14. What does that mean? Well now we have a triangle, where we know all 3 angles and 1 side --> we can find another side by the law of sines:
If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of sines states:
a÷sinA = b÷sinB = c÷sinC
We really need the hypotenuse to then find our hill height, so we'll make hypotenuse = side b in attached image. That being so, then its opposite angle (B) = 141. And the top right angle (C) = 14 with its opposite side (c) = 350ft.
Now we only need b÷sinB = c÷sinC
So b/sin141 = 350/sin14 --> b = 350sin141/sin14 = 350×.63/
b = 220.3/.24 = 910.47 ft
Now that's our hypotenuse, so using our original large right triangle, we can use right triangular trig. to solve. Let's make the right side, our hill height, equal to x.
Sin ¥ = opp. side / hypotenuse -->
Sin ¥ = x / hypotenuse
Sin25 = x / 910.5
x = 910.5×sin25 = 910.5×.423
x = 384.8 ft
So we start by making a large right triangle, because the hill is vertical with the horizontal ground, with 25° as the left angle (where John looks up to top), 90° is the right angle (where ground meets hill base). So we see that the top side of the triangle is the hypotenuse, and equals the line of sight from John to hilltop.
Now we've got additional information that if John walks 350ft towards the hill, his angle of elevation increases by 14. So that = 25+14 = 39. How does that possibly help us?? Well now we can make 2 triangles inside of the one we've already made. So that now we have the triangle base split between the left angle of 25° and where he stopped 350ft to the right of that.
Now the supplement of 39 is 141, and the remaining piece of that too left triangle = 180-25-141 = 14. What does that mean? Well now we have a triangle, where we know all 3 angles and 1 side --> we can find another side by the law of sines:
If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of sines states:
a÷sinA = b÷sinB = c÷sinC
We really need the hypotenuse to then find our hill height, so we'll make hypotenuse = side b in attached image. That being so, then its opposite angle (B) = 141. And the top right angle (C) = 14 with its opposite side (c) = 350ft.
Now we only need b÷sinB = c÷sinC
So b/sin141 = 350/sin14 --> b = 350sin141/sin14 = 350×.63/
b = 220.3/.24 = 910.47 ft
Now that's our hypotenuse, so using our original large right triangle, we can use right triangular trig. to solve. Let's make the right side, our hill height, equal to x.
Sin ¥ = opp. side / hypotenuse -->
Sin ¥ = x / hypotenuse
Sin25 = x / 910.5
x = 910.5×sin25 = 910.5×.423
x = 384.8 ft
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Answer:
The side s has a length of 4 and side q has a length of 4 ⇒ F
Step-by-step explanation:
In the 30°-60°-90° triangle, there is a ratio between its sides
side opp (30°) : side opp (60°) : hypotenuse
1 : : 2
In the given triangle
∵ The side opposite to 30° is s
∵ The side opposite to 60° is q
∵ The hypotenuse is 8
→ Use the ratio above to find the lengths of s and q
side opp (30°) : side opp (60°) : hypotenuse
1 : : 2
s : q : 8
→ By using cross multiplication
∵ s × 2 = 1 × 8
∴ 2s = 8
→ Divide both sides by 2
∴ s = 4
∴ The length of s is 4
∵ q × 2 = × 8
∴ 2q = 8
→ Divide both sides by 2
∴ q = 4
∴ The length of q is 4
a. To solve the first part, we are going to use the formula for the surface area of a sphere: 
where
is the surface area of the sphere
is the radius of the sphere
We know from our problem that
; so lets replace that value in our formula:
To solve the second part, we are going to use the formula for the volume of a sphere:
Where
is the volume of the sphere
is the radius
We know form our problem that, so lets replace that in our formula:
We can conclude that the surface area of the sphere is 314.16 square feet and its volume is 523.6 cubic feet.
b. To solve the first part, we are going to use the formula for the surface area of a square pyramid:
where
is the surface area
is the measure of the base
is the height of the pyramid
We know form our problem that
and 
, so lets replace those value sin our formula:
To solve the second part, we are going to use the formula for the volume of a square pyramid:
where
is the volume
is the measure of the base
is the height of the pyramid
We know form our problem that and , so lets replace those value sin our formula:
We can conclude that the surface area of our pyramid is 266.39 square feet and its volume is 256 cubic feet.
c. To solve the first part, we are going to use the formula for the surface area of a circular cone:
where
is the surface area
is the radius of the circular base
is the height of the cone
We know form our problem that and 
, so lets replace those values in our formula:
![A= \pi (5ft)[(5ft)+ \sqrt{(8ft)^2+(5ft)^2}]](https://tex.z-dn.net/?f=A%3D%20%5Cpi%20%285ft%29%5B%285ft%29%2B%20%5Csqrt%7B%288ft%29%5E2%2B%285ft%29%5E2%7D%5D)
To solve the second part, we are going to use the formula for the volume os a circular cone:
where
is the volume
is the radius of the circular base
is the height of the cone
We know form our problem that and , so lets replace those values in our formula:
We can conclude that the surface area of our cone is 226.73 square feet and its surface area is 209.44 cubic feet.
d. To solve the first part, we are going to use the formula for the surface area of a rectangular prism:
where
is the surface area
is the width
is the length
is the height
We know from our problem that
, 
, and 
, so lets replace those values in our formula:
![A=2[(6ft)(10ft)+(16ft)(10ft)+(16ft)(6ft)]](https://tex.z-dn.net/?f=A%3D2%5B%286ft%29%2810ft%29%2B%2816ft%29%2810ft%29%2B%2816ft%29%286ft%29%5D%20)
To solve the second part, we are going to use the formula for the volume of a rectangular prism:
where
is the volume
is the width
is the length
is the height
We know from our problem that, , and , so lets replace those values in our formula:
We can conclude that the surface area of our solid is 632 square feet and its volume is 960 cubic feet.
e. Remember that a face of a polygon is a side of polygon.
- A sphere has no faces.
- A square pyramid has 5 faces.
- A cone has 1 face.
- A rectangular prism has 6 faces.
Total faces: 5 + 1 + 6 = 12 faces
<span>We can conclude that there are 12 faces in on the four geometric shapes on the holes.
</span>
f. Remember that an edge is a line segment on the boundary of the polygon.
- A sphere has no edges.
- A cone has no edges.
- A rectangular pyramid has 8 edges.
- A rectangular prism has 12 edges.
Total edges: 8 + 20 = 20 edges
Since we have 20 edges in total, we can conclude that your boss will need 20 brackets on the four shapes.
g. Remember that the vertices are the corner points of a polygon.
- A sphere has no vertices.
- A cone has no vertices.
- A rectangular pyramid has 5 vertices.
- A rectangular prism has 8 vertices.
Total vertices: 5 + 8 = 13 vertices
We can conclude that there are 0 vertices for the sphere and the cone; there are 5 vertices for the pyramid, and there are are 8 vertices for the solid (rectangular prism). We can also conclude that your boss will need 13 brackets for the vertices of the four figures.
where
We know from our problem that
To solve the second part, we are going to use the formula for the volume of a sphere:
Where
We know form our problem that
We can conclude that the surface area of the sphere is 314.16 square feet and its volume is 523.6 cubic feet.
b. To solve the first part, we are going to use the formula for the surface area of a square pyramid:
where
We know form our problem that
To solve the second part, we are going to use the formula for the volume of a square pyramid:
where
We know form our problem that
We can conclude that the surface area of our pyramid is 266.39 square feet and its volume is 256 cubic feet.
c. To solve the first part, we are going to use the formula for the surface area of a circular cone:
where
We know form our problem that
To solve the second part, we are going to use the formula for the volume os a circular cone:
where
We know form our problem that
We can conclude that the surface area of our cone is 226.73 square feet and its surface area is 209.44 cubic feet.
d. To solve the first part, we are going to use the formula for the surface area of a rectangular prism:
where
We know from our problem that
To solve the second part, we are going to use the formula for the volume of a rectangular prism:
where
We know from our problem that
We can conclude that the surface area of our solid is 632 square feet and its volume is 960 cubic feet.
e. Remember that a face of a polygon is a side of polygon.
- A sphere has no faces.
- A square pyramid has 5 faces.
- A cone has 1 face.
- A rectangular prism has 6 faces.
Total faces: 5 + 1 + 6 = 12 faces
<span>We can conclude that there are 12 faces in on the four geometric shapes on the holes.
</span>
f. Remember that an edge is a line segment on the boundary of the polygon.
- A sphere has no edges.
- A cone has no edges.
- A rectangular pyramid has 8 edges.
- A rectangular prism has 12 edges.
Total edges: 8 + 20 = 20 edges
Since we have 20 edges in total, we can conclude that your boss will need 20 brackets on the four shapes.
g. Remember that the vertices are the corner points of a polygon.
- A sphere has no vertices.
- A cone has no vertices.
- A rectangular pyramid has 5 vertices.
- A rectangular prism has 8 vertices.
Total vertices: 5 + 8 = 13 vertices
We can conclude that there are 0 vertices for the sphere and the cone; there are 5 vertices for the pyramid, and there are are 8 vertices for the solid (rectangular prism). We can also conclude that your boss will need 13 brackets for the vertices of the four figures.