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snow_tiger [21]

3 years ago

5

The rectangular prism shown has a length of 10 cm, a width of 10 cm, and a height of 6 cm. Find the length of the inner diagonal

AD of the prism. Note that AD is the hypotenuse of right triangle ACD. (Round your answer to the nearest hundredth.)

1 answer:

melamori03 [73]3 years ago

8 0

<h3>Answer:</h3>

15.36 cm

<h3>Explanation:</h3>

The Pythagorean theorem tells you ...

... AC² = AB² + BC² . . . . . relation for face diagonal

... AD² = AC² + CD² . . . . . relation for space diagonal

Subsituting the expression for AC² given by the first equation, the second equaiton becomes ...

... AD² = AB² + BC² + CD²

... AD² = (100 cm²) + (100 cm²) + (36 cm²) = 236 cm²

Taking the square root, we have ...

... AD = √236 cm = 2√59 cm

... AD ≈ 15.36 cm

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Answer: $\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{3arcsin(a) + 2a(1 - a^2)^\Big{\frac{3}{2}} + 3a\sqrt{1 - a^2}}{8} - \frac{3arcsin(b) + 2b(1 - b^2)^\Big{\frac{3}{2}} + 3b\sqrt{1 - b^2}}{8}$ General Formulas and Concepts:

<u>Pre-Calculus</u>

Trigonometric Identities

<u>Calculus</u>

Differentiation

Derivatives
Derivative Notation

Integration

Integrals
Definite/Indefinite Integrals
Integration Constant C

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

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

U-Substitution

Trigonometric Substitution

Reduction Formula: $\displaystyle \int {cos^n(x)} \, dx = \frac{n - 1}{n}\int {cos^{n - 2}(x)} \, dx + \frac{cos^{n - 1}(x)sin(x)}{n}$

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

$\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx$

<u>Step 2: Integrate Pt. 1</u>

<em>Identify variables for u-substitution (trigonometric substitution).</em>

Set <em>u</em>: $\displaystyle x = sin(u)$
[<em>u</em>] Differentiate [Trigonometric Differentiation]: $\displaystyle dx = cos(u) \ du$
Rewrite <em>u</em>: $\displaystyle u = arcsin(x)$

<u>Step 3: Integrate Pt. 2</u>

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[Integrand] Rewrite: $\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \int\limits^a_b {cos(u)[cos^2(u)]^\Big{\frac{3}{2}} \, du$
[Integrand] Simplify: $\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \int\limits^a_b {cos^4(u)} \, du$
[Integral] Reduction Formula: $\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{4 - 1}{4}\int \limits^a_b {cos^{4 - 2}(x)} \, dx + \frac{cos^{4 - 1}(u)sin(u)}{4} \bigg| \limits^a_b$
[Integral] Simplify: $\displaystyle \int\limits^a_b {(1 - x^2)^\Big{\frac{3}{2}}} \, dx = \frac{cos^3(u)sin(u)}{4} \bigg| \limits^a_b + \frac{3}{4}\int\limits^a_b {cos^2(u)} \, du$
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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e

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