A cube's edge lengths are each 5 inches. A rectangular prism's length and width are also 5 inches, but its height is 7

Mathematics

1 answer:

algol [13]2 years ago

6 0

The rectangular prism is 187.5 - 125 = 62.5 cu.inches greater than the volume of cube.

<h3>What is Volume?</h3>

Volume is defined as the space occupied by a three dimensional figure.

It is given that

A cube's edge lengths are each 5 inches

A rectangular prism's length and width are also 5 inches, but its height is $\rm 7\dfrac{1}{2}$

The volume of the cube is given by side * side * side

Volume of cube = 5 *5 * 5 = 125 cu. inches

The volume of cuboid = length * width * height

<h3>Volume of Cuboid = 5 * 5 * 15/2</h3>

Volume of Cuboid = 187.5 cu.inches.

Therefore the rectangular prism is 187.5 - 125 = 62.5 cu.inches greater than the volume of cube.

To know more about Volume

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A. Evaluate ∫20 tan 2x sec^2 2x dx using the substitution u = tan 2x.

irakobra [83]

Answer:

The integral is equal to $5\sec^2(2x)+C$ for an arbitrary constant C.

Step-by-step explanation:

a) If $u=\tan(2x)$ then $du=2\sec^2(2x)dx$ so the integral becomes $\int 20\tan(2x)\sec^2(2x)dx=\int 10\tan(2x) (2\sec^2(2x))dx=\int 10udu=\frac{u^2}{2}+C=10(\int udu)=10(\frac{u^2}{2}+C)=5\tan^2(2x)+C$ . (the constant of integration is actually 5C, but this doesn't affect the result when taking derivatives, so we still denote it by C)

b) In this case $u=\sec(2x)$ hence $du=2\tan(2x)\sec(2x)dx$ . We rewrite the integral as $\int 20\tan(2x)\sec^2(2x)dx=\int 10\sec(2x) (2\tan(2x)\sec(2x))dx=\int 10udu=5\frac{u^2}{2}+C=5\sec^2(2x)+C$ .

c) We use the trigonometric identity $\tan(2x)^2+1=\sec(2x)^2$ is part b). The value of the integral is $5\sec^2(2x)+C=5(\tan^2(2x)+1)+C=5\tan^2(2x)+5+C=5\tan^2(2x)+C$ . which coincides with part a)

Note that we just replaced 5+C by C. This is because we are asked for an indefinite integral. Each value of C defines a unique antiderivative, but we are not interested in specific values of C as this integral is the family of all antiderivatives. Part a) and b) don't coincide for specific values of C (they would if we were working with a definite integral), but they do represent the same family of functions.

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3 years ago

5x-6=3x<br> what does x equal

makvit [3.9K]

5x-6= 3x
⇒ 5x-3x= 6
⇒ 2x= 6
⇒ x= 6/2
⇒ x= 3

Final answer: x=3~

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