Find the minimal length of a right triangle with altitude 1
I have this right triangle here.
The question says: "Suppose we have a right triangle $ABC$, where the right angle is at $C$. Draw the altitude from vertex C to hypotenuse $AB$. If the length of this altitude is $1$ cm, what is the minimal length of the hypotenuse?"
Can someone help me set up the optimization problem? I'm honestly having a hard time setting this up and I'm genuinely stuck. I am given a hint which says "denote by $x$ one of the acute angles".
calculus optimization triangle
asked Nov 20 at 2:06
Future Math person
972717
add a comment |
I have this right triangle here.
The question says: "Suppose we have a right triangle $ABC$, where the right angle is at $C$. Draw the altitude from vertex C to hypotenuse $AB$. If the length of this altitude is $1$ cm, what is the minimal length of the hypotenuse?"
Can someone help me set up the optimization problem? I'm honestly having a hard time setting this up and I'm genuinely stuck. I am given a hint which says "denote by $x$ one of the acute angles".
calculus optimization triangle
asked Nov 20 at 2:06
Future Math person
972717
add a comment |
I have this right triangle here.
The question says: "Suppose we have a right triangle $ABC$, where the right angle is at $C$. Draw the altitude from vertex C to hypotenuse $AB$. If the length of this altitude is $1$ cm, what is the minimal length of the hypotenuse?"
Can someone help me set up the optimization problem? I'm honestly having a hard time setting this up and I'm genuinely stuck. I am given a hint which says "denote by $x$ one of the acute angles".
calculus optimization triangle
asked Nov 20 at 2:06
Future Math person
972717
I have this right triangle here.
The question says: "Suppose we have a right triangle $ABC$, where the right angle is at $C$. Draw the altitude from vertex C to hypotenuse $AB$. If the length of this altitude is $1$ cm, what is the minimal length of the hypotenuse?"
Can someone help me set up the optimization problem? I'm honestly having a hard time setting this up and I'm genuinely stuck. I am given a hint which says "denote by $x$ one of the acute angles".
calculus optimization triangle
calculus optimization triangle
asked Nov 20 at 2:06
Future Math person
972717
asked Nov 20 at 2:06
Future Math person
972717
asked Nov 20 at 2:06
Future Math person
972717
asked Nov 20 at 2:06
Future Math person
972717
asked Nov 20 at 2:06
Future Math person
972717
972717
add a comment |
add a comment |
1 Answer
1
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oldest
votes
Let $D$ be the point of intersection between the drawn altitude an AB. Then,
$$AB=AD+DB$$
From trigonometry,
$$AD=cot x$$
From the geometry of the problem, angle $DCB$ is also $x$, so:
$$DB=tan x$$
All that is left is to minimize,
$$AB=cot x+tan x$$
Subject to $0 leq x leq frac{pi}{2}$.
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Let $D$ be the point of intersection between the drawn altitude an AB. Then,
$$AB=AD+DB$$
From trigonometry,
$$AD=cot x$$
From the geometry of the problem, angle $DCB$ is also $x$, so:
$$DB=tan x$$
All that is left is to minimize,
$$AB=cot x+tan x$$
Subject to $0 leq x leq frac{pi}{2}$.
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
add a comment |
Let $D$ be the point of intersection between the drawn altitude an AB. Then,
$$AB=AD+DB$$
From trigonometry,
$$AD=cot x$$
From the geometry of the problem, angle $DCB$ is also $x$, so:
$$DB=tan x$$
All that is left is to minimize,
$$AB=cot x+tan x$$
Subject to $0 leq x leq frac{pi}{2}$.
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
add a comment |
Let $D$ be the point of intersection between the drawn altitude an AB. Then,
$$AB=AD+DB$$
From trigonometry,
$$AD=cot x$$
From the geometry of the problem, angle $DCB$ is also $x$, so:
$$DB=tan x$$
All that is left is to minimize,
$$AB=cot x+tan x$$
Subject to $0 leq x leq frac{pi}{2}$.
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
Let $D$ be the point of intersection between the drawn altitude an AB. Then,
$$AB=AD+DB$$
From trigonometry,
$$AD=cot x$$
From the geometry of the problem, angle $DCB$ is also $x$, so:
$$DB=tan x$$
All that is left is to minimize,
$$AB=cot x+tan x$$
Subject to $0 leq x leq frac{pi}{2}$.
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
answered Nov 20 at 2:20
Ahmed S. Attaalla
14.7k12049
14.7k12049
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
add a comment |
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
Thank you. I figured out the rest of it :) .
– Future Math person
Nov 20 at 2:37
add a comment |
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