Answer:
<6 = 60 degrees, <7 is 50 degrees, <8 is 70 degrees
Step-by-step explanation:
Angle 6 is 60 degrees because of the alternate interior angles theorem.
Angle 7 is 50 degrees, also because of the alternate interior angles theorem.
Now, the three interior angles of a triangle should add up to 180 degrees. That means:
60 + 50 + <8 = 180
<8 = 70 degrees
<6 = 60
<7 = 50
<8 = 70
If she does 10 a day she will do 520 after 52 weeks
The second function has a graph that is identical to that of the first function, except that it is shifted upward by 4 units.
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The thrust of this question is to get you to recognize that adding a constant to a function shifts its graph in the amount and direction of the constant value (positive ⇒ upward).
The region in question is the set
![R = \left\{ (x, y) : 0 \le x \le 1 \text{ and } x^3 \le y \le 1 \right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%20%28x%2C%20y%29%20%3A%200%20%5Cle%20x%20%5Cle%201%20%5Ctext%7B%20and%20%7D%20x%5E3%20%5Cle%20y%20%5Cle%201%20%5Cright%5C%7D)
or equivalently,
![R = \left\{ (x, y) : 0 \le y \le 1 \text{ and } 0 \le x \le \sqrt[3]{y} \right\}](https://tex.z-dn.net/?f=R%20%3D%20%5Cleft%5C%7B%20%28x%2C%20y%29%20%3A%200%20%5Cle%20y%20%5Cle%201%20%5Ctext%7B%20and%20%7D%200%20%5Cle%20x%20%5Cle%20%5Csqrt%5B3%5D%7By%7D%20%5Cright%5C%7D)
Cross sections are taken perpendicular to the y-axis, which means each section has a base length equal to the horizontal distance between the curve y = x³ and the line x = 0 (the y-axis). This horizontal distance is given by
y = x³ ⇒ x = ∛y
so that each triangular cross section has side length ∛y.
The area of an equilateral triangle with side length s is √3/4 s², so each cross section contributes an infinitesimal area of √3/4 ∛(y²).
Then the volume of this solid is
![\displaystyle \frac{\sqrt3}4 \int_0^1 \sqrt[3]{y^2} \, dy = \frac{\sqrt3}4 \int_0^1 y^{2/3} \, dy = \frac{\sqrt3}4\cdot\frac35 y^{5/3} \bigg|_0^1 = \boxed{\frac{3\sqrt3}{20}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Csqrt3%7D4%20%5Cint_0%5E1%20%5Csqrt%5B3%5D%7By%5E2%7D%20%5C%2C%20dy%20%3D%20%5Cfrac%7B%5Csqrt3%7D4%20%5Cint_0%5E1%20y%5E%7B2%2F3%7D%20%5C%2C%20dy%20%3D%20%5Cfrac%7B%5Csqrt3%7D4%5Ccdot%5Cfrac35%20y%5E%7B5%2F3%7D%20%5Cbigg%7C_0%5E1%20%3D%20%5Cboxed%7B%5Cfrac%7B3%5Csqrt3%7D%7B20%7D%7D)
I've attached some sketches of the solid with 16 and 64 such cross sections to give an idea of what this solid looks like.