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3 years ago

Pretest 5. In a group of 250 students, 40 are seniors. What percentage of the group are seniors?

Mathematics

1 answer:

zvonat [6]3 years ago

8 0

Answer:

$\%seniors = 16\%$

Step-by-step explanation:

Given

$Students = 250$

$Seniors = 40$

Required

Determine the percentage of seniors

This is calculated as thus;

$\%seniors = \frac{Seniors}{Students} * 100\%$

$\%seniors = \frac{40}{250} * 100\%$

$\%seniors = \frac{4000\%}{250}$

$\%seniors = 16\%$

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Step-by-step explanation:

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2 years ago

Look at this system of equations.<br> 2x +5 =9.5<br> 3x + 5y =10.5<br> What is the value of x+y

stiv31 [10]

Answer:

x = 2.25 y = 0.75

Step-by-step explanation:

2x = 9.5- 5

2x = 4.5

x = 4.5 ÷ 2

x = 2.25

know we now what is x so in the second one we evaluate x

3(2.25) + 5y = 10.5

6.75 + 5y = 10.5

5y = 10.5- 6.75

5y = 3.75

y = 3.75 ÷ 5

y = 0.75

that's it !

if you want to check your answer you can evaluate x and y so it will be like

2×2.25 + 5

= 4.5 + 5 = 9.5

and

3× 2.25 +5×0.75

= 6.75 + 3.75 = 10.5

6 0

3 years ago

Prove or disprove (from i=0 to n) sum([2i]^4) <= (4n)^4. If true use induction, else give the smallest value of n that it doe

ddd [48]

Answer:

The statement is true for every n between 0 and 77 and it is false for $n\geq 78$

Step-by-step explanation:

First, observe that, for n=0 and n=1 the statement is true:

For n=0: $\sum^{n}_{i=0} (2i)^4=0 \leq 0=(4n)^4$

For n=1: $\sum^{n}_{i=0} (2i)^4=16 \leq 256=(4n)^4$

From this point we will assume that $n\geq 2$

As we can see, $\sum^{n}_{i=0} (2i)^4=\sum^{n}_{i=0} 16i^4=16\sum^{n}_{i=0} i^4$ and $(4n)^4=256n^4$ . Then,

$\sum^{n}_{i=0} (2i)^4 \leq(4n)^4 \iff \sum^{n}_{i=0} i^4 \leq 16n^4$

Now, we will use the formula for the sum of the first 4th powers:

$\sum^{n}_{i=0} i^4=\frac{n^5}{5} +\frac{n^4}{2} +\frac{n^3}{3}-\frac{n}{30}=\frac{6n^5+15n^4+10n^3-n}{30}$

Therefore:

$\sum^{n}_{i=0} i^4 \leq 16n^4 \iff \frac{6n^5+15n^4+10n^3-n}{30} \leq 16n^4 \\\\ \iff 6n^5+10n^3-n \leq 465n^4 \iff 465n^4-6n^5-10n^3+n\geq 0$

and, because $n \geq 0$ ,

$465n^4-6n^5-10n^3+n\geq 0 \iff n(465n^3-6n^4-10n^2+1)\geq 0 \\\iff 465n^3-6n^4-10n^2+1\geq 0 \iff 465n^3-6n^4-10n^2\geq -1\\\iff n^2(465n-6n^2-10)\geq -1$

Observe that, because $n \geq 2$ and is an integer,

$n^2(465n-6n^2-10)\geq -1 \iff 465n-6n^2-10 \geq 0 \iff n(465-6n) \geq 10\\\iff 465-6n \geq 0 \iff n \leq \frac{465}{6}=\frac{155}{2}=77.5$

In concusion, the statement is true if and only if n is a non negative integer such that $n\leq 77$

So, 78 is the smallest value of n that does not satisfy the inequality.

Note: If you compute $(4n)^4- \sum^{n}_{i=0} (2i)^4$ for 77 and 78 you will obtain:

$(4n)^4- \sum^{n}_{i=0} (2i)^4=53810064$

$(4n)^4- \sum^{n}_{i=0} (2i)^4=-61754992$

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3 years ago

Henry measured a line to be 7.8 inches long. If the actual length of the line is 7.4 inches, then what was the percent error of

Margarita [4]

The percent error of the measurement is 5.4%

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2 years ago

Identify the slope and y intercept of the line having equation y=-8x-15

Harrizon [31]

Answer: the slope would be a negative slope with -15 as the y intercept.

Step-by-step explanation:

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2 years ago

Pretest 5. In a group of 250 students, 40 are seniors. What percentage of the group are seniors?​

Pretest 5. In a group of 250 students, 40 are seniors. What percentage of the group are seniors?