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3 years ago

A 25 foot ladder is leaning against a tree. If the base of the ladder is 6 feet

Mathematics

1 answer:

Vera_Pavlovna [14]3 years ago

4 0

9514 1404 393

Answer:

24.3 ft

Step-by-step explanation:

If h is the height of the ladder up the tree, the geometry can be modeled as a right triangle with legs h and 6, and hypotenuse 25. The Pythagorean theorem gives you the relation ...

h² +6² = 25²

h² = 625 -36 = 589

h = √589 ≈ 24.3

The ladder reaches about 24.3 feet up the tree.

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Determine formula of the nth term 2, 6, 12 20 30,42

nalin [4]

Check the forward differences of the sequence.

If $\{a_n\} = \{2,6,12,20,30,42,\ldots\}$ , then let $\{b_n\}$ be the sequence of first-order differences of $\{a_n\}$ . That is, for n ≥ 1,

$b_n = a_{n+1} - a_n$

so that $\{b_n\} = \{4, 6, 8, 10, 12, \ldots\}$ .

Let $\{c_n\}$ be the sequence of differences of $\{b_n\}$ ,

$c_n = b_{n+1} - b_n$

and we see that this is a constant sequence, $\{c_n\} = \{2, 2, 2, 2, \ldots\}$ . In other words, $\{b_n\}$ is an arithmetic sequence with common difference between terms of 2. That is,

$2 = b_{n+1} - b_n \implies b_{n+1} = b_n + 2$

and we can solve for $b_n$ in terms of $b_1=4$ :

$b_{n+1} = b_n + 2$

$b_{n+1} = (b_{n-1}+2) + 2 = b_{n-1} + 2\times2$

$b_{n+1} = (b_{n-2}+2) + 2\times2 = b_{n-2} + 3\times2$

and so on down to

$b_{n+1} = b_1 + 2n \implies b_{n+1} = 2n + 4 \implies b_n = 2(n-1)+4 = 2(n + 1)$

We solve for $a_n$ in the same way.

$2(n+1) = a_{n+1} - a_n \implies a_{n+1} = a_n + 2(n + 1)$

Then

$a_{n+1} = (a_{n-1} + 2n) + 2(n+1) \\ ~~~~~~~= a_{n-1} + 2 ((n+1) + n)$

$a_{n+1} = (a_{n-2} + 2(n-1)) + 2((n+1)+n) \\ ~~~~~~~ = a_{n-2} + 2 ((n+1) + n + (n-1))$

$a_{n+1} = (a_{n-3} + 2(n-2)) + 2((n+1)+n+(n-1)) \\ ~~~~~~~= a_{n-3} + 2 ((n+1) + n + (n-1) + (n-2))$

and so on down to

$a_{n+1} = a_1 + 2 \displaystyle \sum_{k=2}^{n+1} k = 2 + 2 \times \frac{n(n+3)}2$

$\implies a_{n+1} = n^2 + 3n + 2 \implies \boxed{a_n = n^2 + n}$

6 0

2 years ago

You worked as a lifeguard this summer and earned an extra $3,000. You decided to deposit that amount into your savings account.

Nataly [62]

Answer:

The correct answer is $3300 for simple interest and $3312.24 for compound interest.

Step-by-step explanation:

Income as working as a lifeguard = $3000

We deposit the money in a bank which offers 2% interest annually for a period of 5 years.

Case 1 : Calculating simple interest for the given situation.

Amount after 5 years = 3000 + 3000 × 2 × 5 × $\frac{1}{100}$ = $ ( 3000 + 300) = $3300.

Case 2 : Calculating compound interest for the given situation.

Amount after 5 years = 3000 × $(1 + \frac{2}{100} ) ^{5}$ = $ 3312.24.

Thus the amount after 5 years amount simply is $3000 and compoundly is $3312.24

5 0

3 years ago

In the figure above, what is the area of right<br> triangle QRS?

Sauron [17]

Answer:

I believe it's 20

3 0

3 years ago

Read 2 more answers

Match the irrational numbers with their estimated positions on the number line.

IrinaVladis [17]

The sqrt of 19 is between 4.3 and 4.4.
The sqrt of 18 is between 4.2 and 4.3.
The sqrt of 22 is between 4.6 and 4.7.
And finally, the sqrt of 21 is between 4.5 and 4.6.

3 0

3 years ago

Read 2 more answers

En un censo realizado en escuelas de enseñanza media superior de un municipio, participaron 8 escuelas que indicaron la cantidad

Harrizon [31]

Answer:

In a census carried out in upper secondary schools in a municipality, 8 schools participated, indicating the number of students who graduated in a given year. The data is shown in the following graph and according to this, what was the average number of graduated students?

Step-by-step explanation:

This is what it says in english

7 0

3 years ago