Oaquin counted the number of steps on each level as he ran up the stairs of the tower.
1 answer:
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4. To solve this problem, we divide the two expressions step by step:
Here we have inverted the second term since division is just multiplying the inverse of the term.
In this step we factor out the quadratic equation.
Then, we cancel out the like term which is x-1.
We then solve for the final combined expression:
For the restrictions, we just need to prevent the denominators of the two original terms to reach zero since this would make the expression undefined:
Therefore, x should not be equal to 1, -5, or -4.
Comparing these to the choices, we can tell the correct answer.
ANSWER:; 
5. To get the ratio of the volume of the candle to its surface area, we simply divide the two terms with the volume on the numerator and the surface area on the denominator:
We can simplify this expression by factoring out the denominator and cancelling like terms.
We then rationalize the denominator:
![\frac{rh(3r-3 \sqrt{ r^{2} + h^{2} })}{(3r)^{2}-(3 \sqrt{ r^{2} + h^{2} })^{2}}}=\frac{3 r^{2}h -3rh \sqrt{ r^{2} + h^{2} }}{9r^{2} -9 (r^{2} + h^{2} )}=\frac{3rh(r -\sqrt{ r^{2} + h^{2} })}{9[r^{2} -(r^{2} + h^{2} )]}=\frac{rh(r -\sqrt{ r^{2} + h^{2} })}{3[r^{2} -(r^{2} + h^{2} )]}](https://tex.z-dn.net/?f=%5Cfrac%7Brh%283r-3%20%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%7D%29%7D%7B%283r%29%5E%7B2%7D-%283%20%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%7D%29%5E%7B2%7D%7D%7D%3D%5Cfrac%7B3%20r%5E%7B2%7Dh%20-3rh%20%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%7D%7D%7B9r%5E%7B2%7D%20-9%20%28r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%29%7D%3D%5Cfrac%7B3rh%28r%20-%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%7D%29%7D%7B9%5Br%5E%7B2%7D%20-%28r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%29%5D%7D%3D%5Cfrac%7Brh%28r%20-%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%7D%29%7D%7B3%5Br%5E%7B2%7D%20-%28r%5E%7B2%7D%20%2B%20h%5E%7B2%7D%20%29%5D%7D)
Since the height is equal to the length of the radius, we can replace h with r and further simplify the expression:
![\frac{r*r(r -\sqrt{ r^{2} + r^{2} })}{3[r^{2} -(r^{2} + r^{2} )]}=\frac{ r^{2} (r -\sqrt{2 r^{2} })}{3[r^{2} -(2r^{2} )]}=\frac{ r^{2} (r -r\sqrt{2 })}{-3r^{2} }=\frac{r -r\sqrt{2 }}{-3 }=\frac{r(1 -\sqrt{2 })}{-3 }](https://tex.z-dn.net/?f=%5Cfrac%7Br%2Ar%28r%20-%5Csqrt%7B%20r%5E%7B2%7D%20%2B%20r%5E%7B2%7D%20%7D%29%7D%7B3%5Br%5E%7B2%7D%20-%28r%5E%7B2%7D%20%2B%20r%5E%7B2%7D%20%29%5D%7D%3D%5Cfrac%7B%20r%5E%7B2%7D%20%28r%20-%5Csqrt%7B2%20r%5E%7B2%7D%20%7D%29%7D%7B3%5Br%5E%7B2%7D%20-%282r%5E%7B2%7D%20%29%5D%7D%3D%5Cfrac%7B%20r%5E%7B2%7D%20%28r%20-r%5Csqrt%7B2%20%7D%29%7D%7B-3r%5E%7B2%7D%20%7D%3D%5Cfrac%7Br%20-r%5Csqrt%7B2%20%7D%7D%7B-3%20%7D%3D%5Cfrac%7Br%281%20-%5Csqrt%7B2%20%7D%29%7D%7B-3%20%7D)
By examining the choices, we can see one option similar to the answer.
ANSWER:
Here we have inverted the second term since division is just multiplying the inverse of the term.
In this step we factor out the quadratic equation.
Then, we cancel out the like term which is x-1.
We then solve for the final combined expression:
For the restrictions, we just need to prevent the denominators of the two original terms to reach zero since this would make the expression undefined:
Therefore, x should not be equal to 1, -5, or -4.
Comparing these to the choices, we can tell the correct answer.
ANSWER:
5. To get the ratio of the volume of the candle to its surface area, we simply divide the two terms with the volume on the numerator and the surface area on the denominator:
We can simplify this expression by factoring out the denominator and cancelling like terms.
We then rationalize the denominator:
Since the height is equal to the length of the radius, we can replace h with r and further simplify the expression:
By examining the choices, we can see one option similar to the answer.
ANSWER:
Answer:
Step-by-step explanation:
It would be helpful to know the numbers of rows and columns, so we don't have to guess.
Any number of on-line calculators and/or graphing calculators can multiply these matrices for you. Excel and other spreadsheets can do this, too. Forming the 6 sums of 2 products is tedious to do by hand.
For example, the term in the product matrix at row 1 column 2 is the sum of products of the numbers in row 1 of the first matrix and column 2 of the second matrix:
4·1 + 0·1 = 4
_____
The attachment shows the matrix multiplication on a TI-84 calculator.
