4
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ I added the "algebra-precalculus" tag to your post. Cheers! $\endgroup$ – Robert Lewis Dec 10 '18 at 2:47
  • $\begingroup$ 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. $\endgroup$ – Joel Pereira Dec 10 '18 at 2:47
6
$\begingroup$

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

$(d + 4)^2 = 144 + d^2, \tag 3$

$d^2 + 8d + 16 = 144 + d^2, \tag 4$

$8d + 16 = 144 \Longrightarrow 8d = 128 \Longrightarrow d = 16M \Longrightarrow l = 20M. \tag 5$

$\endgroup$
  • 2
    $\begingroup$ 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. $\endgroup$ – Edward Severinsen Dec 10 '18 at 2:49
  • 1
    $\begingroup$ @EdwardSeverinsen: we're all learners, my friend! $\endgroup$ – Robert Lewis Dec 10 '18 at 2:50
  • 6
    $\begingroup$ No, the answer should be "at that angle, the ladder doesn't rest at the wall" ;-) $\endgroup$ – DonQuiKong Dec 10 '18 at 9:51
  • 1
    $\begingroup$ @DonQuiKong I wouldn't get on it, that's for sure! $\endgroup$ – DoctorPenguin Dec 10 '18 at 11:48
13
$\begingroup$

enter image description here

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 $l = x+4 = 16+4 = 20$

$\endgroup$
  • 4
    $\begingroup$ Nice graphic, +1! $\endgroup$ – Robert Lewis Dec 10 '18 at 2:52
  • 2
    $\begingroup$ @RobertLewis Thanks! $\endgroup$ – Key Flex Dec 10 '18 at 2:57
  • 1
    $\begingroup$ Just as a note 4 times greater than the base is ambiguous and could imply it is x * 4 and not x + 4. $\endgroup$ – Felix Guo Dec 10 '18 at 7:36
  • 2
    $\begingroup$ @FelixGuo "4 times greater" isn't ambiguous at all: it can only mean $x\times 4$ and never $x+4$. It was just a mistake, which has now been corrected. $\endgroup$ – David Richerby Dec 10 '18 at 14:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.