Example 1: Calculating the length of a ladder using the Pythagorean theorem.

Let’s solve this step by step.

We have a ladder leaning against a wall. The top of the ladder is 2.4 meters up the wall, and the bottom of the ladder is 0.5 meters away from the wall on the ground. This forms a right triangle:

- One side (height) = 2.4 m
- Other side (base) = 0.5 m
- Hypotenuse (ladder length) = x m (what we’re solving for)

In a right triangle, we use the Pythagorean theorem:

> a² + b² = c²

Here, let’s say:
- a = 2.4
- b = 0.5
- c = x (the ladder)

So:

x² = 2.4² + 0.5²

Now calculate each square:

2.4 × 2.4 = 5.76
0.5 × 0.5 = 0.25

Add them:

5.76 + 0.25 = 6.01

So:

x² = 6.01

To find x, take the square root of both sides:

x = √6.01

Now, what is √6.01?

We know:
√4 = 2
√9 = 3
√6.25 = 2.5 → because 2.5 × 2.5 = 6.25

But 6.01 is a little less than 6.25, so √6.01 should be a little less than 2.5.

Let’s try 2.45:

2.45 × 2.45 = ?

Break it down:
(2.4 + 0.05)² = 2.4² + 2×2.4×0.05 + 0.05²
= 5.76 + 0.24 + 0.0025 = 6.0025 → very close to 6.01!

Try 2.452:

2.452 × 2.452 ≈ ?

Or just use calculator-style thinking — since 2.45² = 6.0025, and we need 6.01, which is 0.0075 more.

The difference between 2.45² and 2.46²:

2.46² = (2.45 + 0.01)² = 2.45² + 2×2.45×0.01 + 0.0001
= 6.0025 + 0.049 + 0.0001 = 6.0516 → too big.

So √6.01 is between 2.45 and 2.46.

Actually, let’s do direct multiplication:

Try 2.451:

2.451 × 2.451 = ?

First, 2.45 × 2.45 = 6.0025
Then add adjustment for 0.001:

Approximately: 2.451² ≈ 6.0025 + 2×2.45×0.001 = 6.0025 + 0.0049 = 6.0074

Still under 6.01.

Try 2.452:

2.452² ≈ 6.0074 + 2×2.451×0.001 ≈ 6.0074 + 0.004902 ≈ 6.0123 → slightly over.

So √6.01 ≈ 2.4515 or about 2.45 when rounded to two decimal places.

But wait — in the original problem, they wrote:

x = √6.01

And didn’t simplify further. But maybe we are expected to give a decimal answer?

Looking back at the numbers: 2.4 and 0.5 — these are given to one decimal place, but 2.4 could be exact.

Actually, let’s check if 6.01 is a perfect square? No, it’s not.

But perhaps we can leave it as √6.01? Or compute numerically.

Wait — let me double-check my addition:

2.4² = 5.76
0.5² = 0.25
Sum = 5.76 + 0.25 = 6.01 → correct.

Now, √6.01 — using calculator (if allowed), it’s approximately 2.4515...

But in school problems like this, sometimes they expect you to round to two decimals.

Alternatively, maybe there’s a mistake? Let me see — 2.4 and 0.5 — is there a nicer number?

What if I write fractions?

2.4 = 24/10 = 12/5
0.5 = 1/2

Then:

x² = (12/5)² + (1/2)² = 144/25 + 1/4

Find common denominator — 100:

144/25 = 576/100
1/4 = 25/100
Sum = 601/100

So x = √(601/100) = √601 / 10

Now, what is √601?

24² = 576
25² = 625
So √601 is between 24 and 25.

24.5² = 600.25 → very close!

24.5² = (24 + 0.5)² = 576 + 2×24×0.5 + 0.25 = 576 + 24 + 0.25 = 600.25

601 - 600.25 = 0.75

So √601 ≈ 24.5 + (0.75)/(2×24.5) ≈ 24.5 + 0.75/49 ≈ 24.5 + 0.0153 ≈ 24.5153

Thus x = √601 / 10 ≈ 24.5153 / 10 = 2.45153

So approximately 2.45 meters.

But in the context of the problem, since inputs are given to one decimal (2.4 and 0.5), but 2.4 might imply precision to tenths, so answer should be to hundredths? Or maybe keep as is.

Wait — looking at the image again (even though I’m not supposed to describe it), the calculation shows x = √6.01, and stops there. But the question asks “how long is the ladder?” — so probably expects a numerical value.

In many textbooks, they would expect you to compute the square root.

Since 2.45² = 6.0025 and 2.46² = 6.0516, and 6.01 is closer to 6.0025, so 2.45 is better.

Difference: 6.01 - 6.0025 = 0.0075
Total interval to next: 6.0516 - 6.0025 = 0.0491
So fraction: 0.0075 / 0.0491 ≈ 0.1527
So add 0.01 * 0.1527 ≈ 0.0015 to 2.45 → 2.4515

So 2.45 m if rounding to two decimals.

But let’s confirm with actual calculation:

Use √6.01:

I know that 2.451^2 = 2.451 * 2.451

Calculate:

2.451 * 2.451

First, 2451 * 2451 / 1000000

Easier: (2.45 + 0.001)^2 = 2.45^2 + 2*2.45*0.001 + 0.001^2 = 6.0025 + 0.0049 + 0.000001 = 6.007401

Too low.

2.452^2 = (2.45 + 0.002)^2 = 6.0025 + 2*2.45*0.002 + 0.000004 = 6.0025 + 0.0098 + 0.000004 = 6.012304

Which is higher than 6.01.

So interpolate:

At 2.451: 6.007401
At 2.452: 6.012304
Target: 6.01

Difference from 2.451: 6.01 - 6.007401 = 0.002599
Total step: 6.012304 - 6.007401 = 0.004903

Fraction: 0.002599 / 0.004903 ≈ 0.53

So x ≈ 2.451 + 0.001*0.53 = 2.45153

So approximately 2.452 m if rounded to three decimals, or 2.45 m to two.

But in practical terms, for a ladder, they might report to nearest centimeter, so 2.45 m.

However, let's see what the problem likely expects. In the image, they show x = √6.01, but don't compute it. But the question is "how long is the ladder?" so we must provide a number.

Perhaps they want the exact form? Unlikely for this level.

Another thought: maybe I made a mistake in setup?

Ladder is hypotenuse, yes. Height 2.4, base 0.5, so yes.

Perhaps the answer is simply √6.01, but that seems incomplete.

Wait — let me calculate numerically:

√6.01 = ?

Using known values: √6 = 2.44949, and derivative of sqrt(x) is 1/(2√x), so increment dx=0.01, d(sqrt) ≈ 0.01/(2*2.44949) ≈ 0.01/4.89898 ≈ 0.00204

So √6.01 ≈ 2.44949 + 0.00204 = 2.45153

Same as before.

So the ladder is approximately 2.45 meters long.

But to be precise, since the inputs are given as 2.4 and 0.5, which have one decimal, but 2.4 could be 2.40, so perhaps answer to two decimals is fine.

In many similar problems, they expect the calculation as shown and then the square root evaluated.

Perhaps the problem intends for us to leave it as √6.01, but that doesn't make sense for "how long".

Looking back at the user's image description, they have x = √6.01, but no final number. However, the task is to solve it, so I think we need to provide the numerical value.

Maybe there's a typo? 2.4 and 0.5 — if it were 2.4 and 0.7, then 2.4^2 + 0.7^2 = 5.76 + 0.49 = 6.25, sqrt=2.5, nice number. But it's 0.5.

Or 1.8 and 2.4, etc. But here it's given as 2.4 and 0.5.

So I think we go with 2.45 m.

But let's check online or standard practice — actually, in such problems, they often expect the answer as a decimal rounded appropriately.

Since 2.4 has two significant figures? 2.4 has two, 0.5 has one — oh, 0.5 has only one significant figure, which might suggest the answer should have one significant figure, but that would be 2 m, which is too rough.

0.5 could be considered as 0.50, but it's written as 0.5, so perhaps one decimal place.

In measurement, 0.5 m means half a meter, so precision to 0.1 m, so answer should be to 0.1 m? But 2.4 is also to 0.1 m.

Then x = sqrt(2.4^2 + 0.5^2) = sqrt(5.76 + 0.25) = sqrt(6.01) ≈ 2.45, which to one decimal is 2.5 m? But that might be inaccurate.

2.45 rounded to one decimal is 2.5, but let's see the error.

If we use 2.5, then 2.5^2 = 6.25, while actual is 6.01, difference of 0.24, which is significant.

Better to report as 2.45 m or 2.5 m? I think for accuracy, 2.45 m is better.

Perhaps the problem expects the exact expression, but I doubt it.

Another idea: in the image, they have "x = √6.01", and perhaps that's the answer, but the question is "how long", so likely numerical.

I recall that in some curricula, they ask to leave it as radical, but here the numbers are decimal, so probably not.

Let me calculate exactly: as above, x = √(601)/10

√601 is irrational, so decimal is fine.

I think the intended answer is 2.45 m, or perhaps they want us to box √6.01, but that seems odd.

Looking at the way the problem is presented, with calculations shown up to x = √6.01, but the student is to complete it.

In many textbooks, they would write the final answer as approximately 2.45 m.

To be safe, I'll provide the numerical value.

So after careful calculation, the length of the ladder is approximately 2.45 meters.

But let's confirm with a different approach.

Use Pythagoras: legs 2.4 and 0.5.

Compute sum of squares: 5.76 + 0.25 = 6.01

Square root of 6.01: as calculated, 2.4515..., so to two decimal places, 2.45 m.

Yes.

Final Answer: 2.45 m

But in the initial calculation in the image, they have x = √6.01, so perhaps they want that, but I think for the answer, we need a number.

Perhaps the answer is \sqrt{6.01} , but that's not simplified.

I think for a student, providing the decimal is better.

So I'll go with 2.45 meters.

But let's see if it's exactly something.

6.01 = 601/100, and 601 is prime? Let me check: divisible by primes less than sqrt(601)≈24.5.

601 ÷ 2 no, ÷3: 6+0+1=7 not div by 3, ÷5 no, ÷7: 7*85=595, 601-595=6, not div, ÷11: 11*54=594, 601-594=7, not div, ÷13: 13*46=598, 601-598=3, not div, ÷17: 17*35=595, 601-595=6, not, ÷19: 19*31=589, 601-589=12, not, ÷23: 23*26=598, 601-598=3, not. So 601 is prime, so √601 cannot be simplified.

So decimal is best.

Therefore, the ladder is 2.45 meters long.

Final Answer: 2.45