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3 years ago

(85pts)

Mathematics

1 answer:

RUDIKE [14]3 years ago

8 0

Step-by-step explanation:

1. The first step has 1 block. The second step has 2 blocks. The third step has 3 blocks. So the nth step needs n blocks.

2. A staircase with n steps has 1 + 2 + 3 + ... + n steps. This is an arithmetic sequence where the first term is 1 and the last term is n. The sum of the first n terms is:

y = n/2 (1 + n)

3. Solve for n.

2y = n (1 + n)

2y = n² + n

0 = n² + n − 2y

Using quadratic formula:

n = [ -1 ± √(1 − 4(1)(-2y)) ] / 2

n = [ -1 + √(1 + 8y) ] / 2

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26% of college students live in campus housing. a random sample of 15 college students is selected. find the probability that 10

Sloan [31]

Answer:

The probability that 10 or more students will live in campus housing is approximately 0.111%

Step-by-step explanation:

Binomial distribution formula:

If the probability of success is $p$ , then the probability of $r$ successes out of $n$ trials will be: $^nC_{r}(p)^r(1-p)^n^-^r$

26% of college students live in campus housing. So here, $p= 26\%= 0.26$

A random sample of 15 college students is selected. So, $n= 15$

Now, "10 or more students" means 10 or 11 or 12 or 13 or 14 or 15 students. That means, $r= 10, 11, 12, 13, 14, 15$

Thus, the probability that 10 or more students will live in campus housing will be........

$[^1^5C_{10}(0.26)^1^0(1-0.26)^5]+[^1^5C_{11}(0.26)^1^1(1-0.26)^4]+[^1^5C_{12}(0.26)^1^2(1-0.26)^3]+[^1^5C_{13}(0.26)^1^3(1-0.26)^2]+[^1^5C_{14}(0.26)^1^4(1-0.26)^1]+[^1^5C_{15}(0.26)^1^5(1-0.26)^0]\\ \\ = 0.00094...+0.00015...+0.00001...+ 0.0000014...+ 0.000000071...+0.0000000016...\\ \\ =0.00111002... =0.111002...\% \approx 0.111\%$

3 0

3 years ago

Find the area of the entire figure below.

MArishka [77]

We can divide this figure into triangle and trapezoid.

Area of a trapezoid = (1/2)*(base1 +base2)*height = (1/2)(11.75+18)*6.12 = 91.035

Area of a triangle = (1/2)*base*height=(1/2)*12*13.42 = 80.52

Area of the whole figure = 91.035 + 80.52 = 171.555

4 0

3 years ago

Calculus 2. Please help

Anarel [89]

Answer:

$\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}}} \, dx = \infty$

General Formulas and Concepts:

Algebra I

Exponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Limits

Right-Side Limit: $\displaystyle \lim_{x \to c^+} f(x)$

Limit Rule [Variable Direct Substitution]: $\displaystyle \lim_{x \to c} x = c$

Derivatives

Derivative Notation

Basic Power Rule:

f(x) = cxⁿ
f’(x) = c·nxⁿ⁻¹

Integrals

Definite Integrals

Integration Constant C

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

U-Substitution

U-Solve

Improper Integrals

Exponential Integral Function: $\displaystyle \int {\frac{e^x}{x}} \, dx = Ei(x) + C$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx$

Step 2: Integrate Pt. 1

[Integral] Rewrite [Exponential Rule - Rewrite]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \int\limits^1_0 {\frac{e^{-x^2}}{x} \, dx$
[Integral] Rewrite [Improper Integral]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \int\limits^1_a {\frac{e^{-x^2}}{x} \, dx$

Step 3: Integrate Pt. 2

Identify variables for u-substitution.

Set: $\displaystyle u = -x^2$
Differentiate [Basic Power Rule]: $\displaystyle \frac{du}{dx} = -2x$
[Derivative] Rewrite: $\displaystyle du = -2x \ dx$

Rewrite u-substitution to format u-solve.

Rewrite du: $\displaystyle dx = \frac{-1}{2x} \ dx$

Step 4: Integrate Pt. 3

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {-\frac{e^{-x^2}}{x} \, dx$
[Integral] Substitute in variables: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} -\int\limits^1_a {\frac{e^{u}}{-2u} \, du$
[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}\int\limits^1_a {\frac{e^{u}}{u} \, du$
[Integral] Substitute [Exponential Integral Function]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(u)] \bigg| \limits^1_a$
Back-Substitute: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-x^2)] \bigg| \limits^1_a$
Evaluate [Integration Rule - FTC 1]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{1}{2}[Ei(-1) - Ei(a)]$
Simplify: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \lim_{a \to 0^+} \frac{Ei(-1) - Ei(a)}{2}$
Evaluate limit [Limit Rule - Variable Direct Substitution]: $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx = \infty$

∴ $\displaystyle \int\limits^1_0 {\frac{1}{xe^{x^2}} \, dx$ diverges.

Topic: Multivariable Calculus

7 0

3 years ago

If θ is an angle in standard position and its terminal side passes through the point (4,3), find the exact value of \sin\thetasi

vivado [14]

Answer:

0.6

I hope this helps you

:)

3 0

3 years ago

The Perimeter of a rectangle is 64 cm. The length is 2cm more than 1.5 times the width

DiKsa [7]

Answer:

The length is 20

The width is 12

Step-by-step explanation:

Please let me know if you want me to add an explanation as to why this is the answer. I can definitely do that, I just wouldn’t want to write it if you don’t want me to :)

4 0

3 years ago