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3 years ago

Please help me with this!

Mathematics

2 answers:

salantis [7]3 years ago

6 0

The original area of a face would be a^2. Now that you added b to the edge, the new area of each face would be (a+b)^2. To find how much the are increased, subtract a^2 from (a+b)^2. $(a+b)^2-(a^2)= a^2+2ab+b^2-(a^2)= 2ab+b^2= b(2a+b)$ So the answer is b(2a+b)

viktelen [127]3 years ago

3 0

Volume of the cube = a³ (Given)

$\boxed{ \text{Volume of a cube = side}^3}$

Side of a cube = ∛a³ = a

Side of the cube after it increased by b = a + b

$\boxed{\text{Area of a cube = side}^2}$

Area of the cube = (a + b)²

Increase in area = (a + b)² - a²

Simplify (a + b)² - a²:

(a + b)² - a²

Open (a + b)² as a² + 2ab + b² :

= a² + 2ab + b² - a²

Combine a² and -a²:

= 2ab + b²

Take out b as the common factor:

= b(2a + b)

Answer: (H) b(2a + b)

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Find the exact length of the curve. 36y2 = (x2 − 4)3, 5 ≤ x ≤ 9, y ≥ 0

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We are looking for the length of a curve, also known as the arc length. Before we get to the formula for arc length, it would help if we re-wrote the equation in y = form.

We are given: $36 y^{2} =( x^{2} -4)^3$
We divide by 36 and take the root of both sides to obtain: $y = \sqrt{ \frac{( x^{2} -4)^3}{36} }$

Note that the square root can be written as an exponent of 1/2 and so we can further simplify the above to obtain: $y = \frac{( x^{2} -4)^{3/2}}{6} }=( \frac{1}{6} )(x^{2} -4)^{3/2}}$

Let's leave that for the moment and look at the formula for arc length. The formula is $L= \int\limits^c_d {ds}$ where ds is defined differently for equations in rectangular form (which is what we have), polar form or parametric form.

Rectangular form is an equation using x and y where one variable is defined in terms of the other. We have y in terms of x. For this, we define ds as follows: $ds= \sqrt{1+( \frac{dy}{dx})^2 } dx$

As a note for a function x in terms of y simply switch each dx in the above to dy and vice versa.

As you can see from the formula we need to find dy/dx and square it. Let's do that now.

We can use the chain rule: bring down the 3/2, keep the parenthesis, raise it to the 3/2 - 1 and then take the derivative of what's inside (here $x^2-4$ ). More formally, we can let $u=x^{2} -4$ and then consider the derivative of $u^{3/2}du$ . Either way, we obtain,

$\frac{dy}{dx}=( \frac{1}{6})( x^{2} -4)^{1/2}(2x)=( \frac{x}{2})( x^{2} -4)^{1/2}$

Looking at the formula for ds you see that dy/dx is squared so let's square the dy/dx we just found.
$( \frac{dy}{dx}^2)=( \frac{x^2}{4})( x^{2} -4)= \frac{x^4-4 x^{2} }{4}$

This means that in our case:
$ds= \sqrt{1+\frac{x^4-4 x^{2} }{4}} dx$
$ds= \sqrt{\frac{4}{4}+\frac{x^4-4 x^{2} }{4}} dx$
$ds= \sqrt{\frac{x^4-4 x^{2}+4 }{4}} dx$
$ds= \sqrt{\frac{( x^{2} -2)^2 }{4}} dx$
$ds= \frac{x^2-2}{2}dx =( \frac{1}{2} x^{2} -1)dx$

Recall, the formula for arc length: $L= \int\limits^c_d {ds}$
Here, the limits of integration are given by 5 and 9 from the initial problem (the values of x over which we are computing the length of the curve). Putting it all together we have:

$L= \int\limits^9_5 { \frac{1}{2} x^{2} -1 } \, dx = (\frac{1}{2}) ( \frac{x^3}{3}) -x$ evaluated from 9 to 5 (I cannot seem to get the notation here but usually it is a straight line with the 9 up top and the 5 on the bottom -- just like the integral with the 9 and 5 but a straight line instead). This means we plug 9 into the expression and from that subtract what we get when we plug 5 into the expression.

That is, $[(\frac{1}{2}) ( \frac{9^3}{3}) -9]-([(\frac{1}{2}) ( \frac{5^3}{3}) -5]=( \frac{9^3}{6}-9)-( \frac{5^3}{6}-5})=\frac{290}{3}$

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3 years ago