How can I deduce the hypotenuse from the information given?
up vote
4
down vote
favorite
I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:
A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.
I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?
calculus algebra-precalculus trigonometry
edited Dec 12 at 4:12
Key Flex
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
4
down vote
favorite
I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:
A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.
I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?
calculus algebra-precalculus trigonometry
edited Dec 12 at 4:12
Key Flex
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:
A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.
I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?
calculus algebra-precalculus trigonometry
edited Dec 12 at 4:12
Key Flex
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I'm going into Machine Learning and am currently brushing up on some Calculus on Coursera. Everything was going smoothly until I got to this word problem:
A ladder rests against a wall. The top of the ladder touches the wall at height $12$ meters. The length of the ladder is $4$ meters longer than the distance from the base of the ladder to the wall. Find the length of the ladder.
I am confused as to how to deduce the hypotenuse from the information given above. And have sat here trying different things with no success. What am I missing?
calculus algebra-precalculus trigonometry
calculus algebra-precalculus trigonometry
edited Dec 12 at 4:12
Key Flex
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 12 at 4:12
Key Flex
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Dec 12 at 4:12
Key Flex
7,41441232
edited Dec 12 at 4:12
Key Flex
7,41441232
edited Dec 12 at 4:12
Key Flex
7,41441232
7,41441232
asked Dec 10 at 2:38
Edward Severinsen
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 10 at 2:38
Edward Severinsen
1233
asked Dec 10 at 2:38
Edward Severinsen
1233
1233
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Edward Severinsen is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47
add a comment |
I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47
I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47
add a comment |
2 Answers
2
active
oldest
votes
up vote
6
down vote
accepted
Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then
$l = d + 4; tag 1$
since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write
$l^2 = (12)^2 + d^2; tag 2$
substituting (1) into (2) yields
$(d + 4)^2 = 144 + d^2, tag 3$
$d^2 + 8d + 16 = 144 + d^2, tag 4$
$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
2
Oh my. I distributed the 2 exponent todand4individually instead of multiplying the expression by itself. Not the first time this has gotten me.
– Edward Severinsen
Dec 10 at 2:49
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
add a comment |
up vote
13
down vote
Given the length of the wall as $12$.
Take the length of the base as $x$.
Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$
Now according to the pythagorean theorem we have,
$begin{align}
(x+4)^2 &= 12^2 + x^2 \
x^2 + 16 + 8x &= 144 + x^2 \
8x &= 128 \
x & = 16
end{align}$
So, the length of the ladder $l = x+4 = 16+4 = 20$
edited Dec 10 at 8:17
SQB
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
Just as a note4 times greater than the baseis ambiguous and could imply it isx * 4and notx + 4.
– Felix Guo
Dec 10 at 7:36
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then
$l = d + 4; tag 1$
since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write
$l^2 = (12)^2 + d^2; tag 2$
substituting (1) into (2) yields
$(d + 4)^2 = 144 + d^2, tag 3$
$d^2 + 8d + 16 = 144 + d^2, tag 4$
$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
2
Oh my. I distributed the 2 exponent todand4individually instead of multiplying the expression by itself. Not the first time this has gotten me.
– Edward Severinsen
Dec 10 at 2:49
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
add a comment |
up vote
6
down vote
accepted
Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then
$l = d + 4; tag 1$
since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write
$l^2 = (12)^2 + d^2; tag 2$
substituting (1) into (2) yields
$(d + 4)^2 = 144 + d^2, tag 3$
$d^2 + 8d + 16 = 144 + d^2, tag 4$
$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
2
Oh my. I distributed the 2 exponent todand4individually instead of multiplying the expression by itself. Not the first time this has gotten me.
– Edward Severinsen
Dec 10 at 2:49
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
add a comment |
up vote
6
down vote
accepted
up vote
6
down vote
accepted
Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then
$l = d + 4; tag 1$
since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write
$l^2 = (12)^2 + d^2; tag 2$
substituting (1) into (2) yields
$(d + 4)^2 = 144 + d^2, tag 3$
$d^2 + 8d + 16 = 144 + d^2, tag 4$
$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
Let $d$ be the distance from the ladder to the wall, and $l$ the length of the ladder.
Then
$l = d + 4; tag 1$
since the wall is mos' likely perpendicular to the ground, we may deploy the Pythagorean theorem and write
$l^2 = (12)^2 + d^2; tag 2$
substituting (1) into (2) yields
$(d + 4)^2 = 144 + d^2, tag 3$
$d^2 + 8d + 16 = 144 + d^2, tag 4$
$8d + 16 = 144 Longrightarrow 8d = 128 Longrightarrow d = 16M Longrightarrow l = 20M. tag 5$
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
answered Dec 10 at 2:46
Robert Lewis
42.8k22863
42.8k22863
2
Oh my. I distributed the 2 exponent todand4individually instead of multiplying the expression by itself. Not the first time this has gotten me.
– Edward Severinsen
Dec 10 at 2:49
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
add a comment |
2
Oh my. I distributed the 2 exponent todand4individually instead of multiplying the expression by itself. Not the first time this has gotten me.
– Edward Severinsen
Dec 10 at 2:49
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
2
2
Oh my. I distributed the 2 exponent to
d and 4 individually instead of multiplying the expression by itself. Not the first time this has gotten me.– Edward Severinsen
Dec 10 at 2:49
Oh my. I distributed the 2 exponent to
d and 4 individually instead of multiplying the expression by itself. Not the first time this has gotten me.– Edward Severinsen
Dec 10 at 2:49
1
1
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
@EdwardSeverinsen: we're all learners, my friend!
– Robert Lewis
Dec 10 at 2:50
6
6
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-)
– DonQuiKong
Dec 10 at 9:51
1
1
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
@DonQuiKong I wouldn't get on it, that's for sure!
– JTPenguin
Dec 10 at 11:48
add a comment |
up vote
13
down vote
Given the length of the wall as $12$.
Take the length of the base as $x$.
Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$
Now according to the pythagorean theorem we have,
$begin{align}
(x+4)^2 &= 12^2 + x^2 \
x^2 + 16 + 8x &= 144 + x^2 \
8x &= 128 \
x & = 16
end{align}$
So, the length of the ladder $l = x+4 = 16+4 = 20$
edited Dec 10 at 8:17
SQB
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
Just as a note4 times greater than the baseis ambiguous and could imply it isx * 4and notx + 4.
– Felix Guo
Dec 10 at 7:36
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
add a comment |
up vote
13
down vote
Given the length of the wall as $12$.
Take the length of the base as $x$.
Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$
Now according to the pythagorean theorem we have,
$begin{align}
(x+4)^2 &= 12^2 + x^2 \
x^2 + 16 + 8x &= 144 + x^2 \
8x &= 128 \
x & = 16
end{align}$
So, the length of the ladder $l = x+4 = 16+4 = 20$
edited Dec 10 at 8:17
SQB
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
Just as a note4 times greater than the baseis ambiguous and could imply it isx * 4and notx + 4.
– Felix Guo
Dec 10 at 7:36
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
add a comment |
up vote
13
down vote
up vote
13
down vote
Given the length of the wall as $12$.
Take the length of the base as $x$.
Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$
Now according to the pythagorean theorem we have,
$begin{align}
(x+4)^2 &= 12^2 + x^2 \
x^2 + 16 + 8x &= 144 + x^2 \
8x &= 128 \
x & = 16
end{align}$
So, the length of the ladder $l = x+4 = 16+4 = 20$
edited Dec 10 at 8:17
SQB
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
Given the length of the wall as $12$.
Take the length of the base as $x$.
Since the length of the ladder $l$ is $4$ meters greater than the base, we have $l = x+4$
Now according to the pythagorean theorem we have,
$begin{align}
(x+4)^2 &= 12^2 + x^2 \
x^2 + 16 + 8x &= 144 + x^2 \
8x &= 128 \
x & = 16
end{align}$
So, the length of the ladder $l = x+4 = 16+4 = 20$
edited Dec 10 at 8:17
SQB
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
edited Dec 10 at 8:17
SQB
1,73211026
edited Dec 10 at 8:17
SQB
1,73211026
edited Dec 10 at 8:17
SQB
1,73211026
1,73211026
answered Dec 10 at 2:49
Key Flex
7,41441232
answered Dec 10 at 2:49
Key Flex
7,41441232
answered Dec 10 at 2:49
Key Flex
7,41441232
7,41441232
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
Just as a note4 times greater than the baseis ambiguous and could imply it isx * 4and notx + 4.
– Felix Guo
Dec 10 at 7:36
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
add a comment |
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
Just as a note4 times greater than the baseis ambiguous and could imply it isx * 4and notx + 4.
– Felix Guo
Dec 10 at 7:36
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
4
4
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
Nice graphic, +1!
– Robert Lewis
Dec 10 at 2:52
2
2
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
@RobertLewis Thanks!
– Key Flex
Dec 10 at 2:57
1
1
Just as a note
4 times greater than the base is ambiguous and could imply it is x * 4 and not x + 4.– Felix Guo
Dec 10 at 7:36
Just as a note
4 times greater than the base is ambiguous and could imply it is x * 4 and not x + 4.– Felix Guo
Dec 10 at 7:36
2
2
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
@FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $xtimes 4$ and never $x+4$. It was just a mistake, which has now been corrected.
– David Richerby
Dec 10 at 14:47
add a comment |
Edward Severinsen is a new contributor. Be nice, and check out our Code of Conduct.
Edward Severinsen is a new contributor. Be nice, and check out our Code of Conduct.
Edward Severinsen is a new contributor. Be nice, and check out our Code of Conduct.
Edward Severinsen is a new contributor. Be nice, and check out our Code of Conduct.
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I added the "algebra-precalculus" tag to your post. Cheers!
– Robert Lewis
Dec 10 at 2:47
Draw a picture with the ladder. Label the sides of the triangle formed by the ladder, the wall, and the floor. If you call the base x, the ladder is x+4. Then use the Pythagorean theorem.
– Joel Pereira
Dec 10 at 2:47