<h3>
Answer: 2 < x < 7</h3>
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Explanation:
We'll use the hinge theorem here. This says (more or less) that the larger an angle is, the side opposite of that will be longer.
Imagine that the segments with the single tickmarks represent a door swinging open/shut. The more open a door is, the further the distance it is from the handle to the frame. In terms of these triangles, the segment 25 is larger than 5x-10 because the angle 90 (opposite the 25) is larger than the angle 85 degrees (opposite the 5x-10)
In symbols we say
5x - 10 < 25
We also say 5x - 10 > 0 or 0 < 5x - 10 to ensure that the segment of length 5x-10 is not 0 or negative.
Put the two inequalities together and we get 0 < 5x - 10 < 25
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Solve for x
0 < 5x - 10 < 25
0+10 < 5x - 10 < 25 + 10 .... adding 10 to all sides
10 < 5x < 35
10/5 < 5x/5 < 35/5 ... dividing all sides by 5
2 < x < 7
We have x between 2 and 7, and not equal to either endpoint.
rationalizing the numerator, or namely, "getting rid of that pesky radical at the top".
we simply multiply top and bottom by a value that will take out the radicand in the numerator.
![\bf \cfrac{\sqrt[3]{144x}}{\sqrt[3]{y}}~~ \begin{cases} 144=2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3\\ \qquad 2^3\cdot 18 \end{cases}\implies \cfrac{\sqrt[3]{2^3\cdot 18x}}{\sqrt[3]{y}}\implies \cfrac{2\sqrt[3]{ 18x}}{\sqrt[3]{y}} \\\\\\ \cfrac{2\sqrt[3]{ 18x}}{\sqrt[3]{y}}\cdot \cfrac{\sqrt[3]{(18x)^2}}{\sqrt[3]{(18x)^2}}\implies \cfrac{2\sqrt[3]{(18x)(18x)^2}}{\sqrt[3]{(y)(18x)^2}}\implies \cfrac{2\sqrt[3]{(18x)^3}}{\sqrt[3]{18^2x^2y}}](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7B%5Csqrt%5B3%5D%7B144x%7D%7D%7B%5Csqrt%5B3%5D%7By%7D%7D~~%0A%5Cbegin%7Bcases%7D%0A144%3D2%5Ccdot%202%5Ccdot%202%5Ccdot%202%5Ccdot%203%5Ccdot%203%5C%5C%0A%5Cqquad%202%5E3%5Ccdot%2018%0A%5Cend%7Bcases%7D%5Cimplies%20%5Ccfrac%7B%5Csqrt%5B3%5D%7B2%5E3%5Ccdot%20%2018x%7D%7D%7B%5Csqrt%5B3%5D%7By%7D%7D%5Cimplies%20%5Ccfrac%7B2%5Csqrt%5B3%5D%7B%20%2018x%7D%7D%7B%5Csqrt%5B3%5D%7By%7D%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7B2%5Csqrt%5B3%5D%7B%20%2018x%7D%7D%7B%5Csqrt%5B3%5D%7By%7D%7D%5Ccdot%20%5Ccfrac%7B%5Csqrt%5B3%5D%7B%2818x%29%5E2%7D%7D%7B%5Csqrt%5B3%5D%7B%2818x%29%5E2%7D%7D%5Cimplies%20%5Ccfrac%7B2%5Csqrt%5B3%5D%7B%2818x%29%2818x%29%5E2%7D%7D%7B%5Csqrt%5B3%5D%7B%28y%29%2818x%29%5E2%7D%7D%5Cimplies%20%5Ccfrac%7B2%5Csqrt%5B3%5D%7B%2818x%29%5E3%7D%7D%7B%5Csqrt%5B3%5D%7B18%5E2x%5E2y%7D%7D)
![\bf \cfrac{2(18x)}{\sqrt[3]{324x^2y}}~~ \begin{cases} 324=2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 3\\ \qquad 12\cdot 3^3 \end{cases}\implies \cfrac{36x}{\sqrt[3]{12\cdot 3^3x^2y}} \\\\\\ \cfrac{36x}{3\sqrt[3]{12x^2y}}\implies \cfrac{12x}{\sqrt[3]{12x^2y}}](https://tex.z-dn.net/?f=%5Cbf%20%5Ccfrac%7B2%2818x%29%7D%7B%5Csqrt%5B3%5D%7B324x%5E2y%7D%7D~~%0A%5Cbegin%7Bcases%7D%0A324%3D2%5Ccdot%202%5Ccdot%203%5Ccdot%203%5Ccdot%203%5Ccdot%203%5C%5C%0A%5Cqquad%2012%5Ccdot%203%5E3%0A%5Cend%7Bcases%7D%5Cimplies%20%5Ccfrac%7B36x%7D%7B%5Csqrt%5B3%5D%7B12%5Ccdot%203%5E3x%5E2y%7D%7D%0A%5C%5C%5C%5C%5C%5C%0A%5Ccfrac%7B36x%7D%7B3%5Csqrt%5B3%5D%7B12x%5E2y%7D%7D%5Cimplies%20%5Ccfrac%7B12x%7D%7B%5Csqrt%5B3%5D%7B12x%5E2y%7D%7D)
9514 1404 393
Answer:
25 +0i
Step-by-step explanation:
The conjugate of a complex number is that number with the sign of the imaginary part reversed.
For z = -3+4i, its conjugate z* is -3-4i. The product of z and z* is ...
(-3 +4i)(-3 -4i) = -3(-3 -4i) +4i(-3 -4i)
= 9 +12i -12i -16i² = 9 +16 = 25
The real part of the product is 25; the imaginary part is 0.
(-3 +4i)(-3 -4i) = 25 +0i
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You may have noticed that (z)(z*) = |z|², the sum of the squares of the real and imaginary parts. It is always a non-negative real number.
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Answer:
The radius of the balloon is increasing at a rate of 4 feet per minute.
Step-by-step explanation:
We are given the following in the question:
Volume of sphere is given by
where r is the radius of the balloon.
Instant radius, r = 2 ft
Rate of change of volume =
Putting values, we get,
Thus, the radius of the balloon is increasing at a rate of 4 feet per minute.
