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3 years ago

Please help AS LEVEL c2 tell me how to work out

Mathematics

1 answer:

Korolek [52]3 years ago

3 0

$\bf [1-tan(\theta)][2sin(\theta)+1]=0\implies 
\begin{cases}
1-tan(\theta)=0\\\\
1=tan(\theta)\\\\
tan^{-1}(1)=\theta\\\\
\qquad \frac{\pi }{4},\frac{5\pi }{4}\\
----------\\
2sin(\theta)+1=0\\\\
2sin(\theta)=-1\\\\
sin(\theta)=\frac{-1}{2}\\\\
sin^{-1}\left(-\frac{1}{2} \right)=\theta\\\\
\qquad \frac{7\pi }{6},\frac{11\pi }{6}
\end{cases}$

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What is the volume of the composite figure? (Round to the nearest hundredth. Use 3.14 for x.)

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The volume of the composite figure is the third option 385.17 cubic centimeters.

Step-by-step explanation:

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The composite figure consists of a cone and a half-sphere on top.

We will have to calculate the volumes of the cone and the half-sphere separately and then add them to obtain the total volume.

Step 2:

The volume of a cone is determined by multiplying $\frac{1}{3}$ with π, the square of the radius (r²) and height (h). Here we substitute π as 3.1415.

The radius is 4 cm and the height is 15 cm.

The volume of the cone :

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Step 3:

The area of a half-sphere is half of a full sphere.

The volume of a sphere is given by multiplying $\frac{4}{3}$ with π and the cube of the radius (r³).

Here the radius is 4 cm. We take π as 3.1415.

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The volume of the half-sphere $=\frac{1}{2} (268.07) = 134.037$ cubic cm.

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3 years ago

Find the dot product of the position vectors whose terminal points are (14, 9) and (3, 6).

erma4kov [3.2K]

Answer:

Step-by-step explanation:

The formula for the dot product of vectors is

u·v = |u||v|cosθ

where |u| and |v| are the magnitudes (lengths) of the vectors. The formula for that is the same as Pythagorean's Theorem.

$|u|=\sqrt{14^2+9^2}$ which is $\sqrt{277}$

$|v|=\sqrt{3^2+6^2}$ which is $\sqrt{45}$

I am assuming by looking at the above that you can determine where the numbers under the square root signs came from. It's pretty apparent.

We also need the angle, which of course has its own formula.

$cos\theta=\frac{uv}{|u||v|}$ where uv has ITS own formula:

uv = (14 * 3) + (9 * 6) which is taking the numbers in the i positions in the first set of parenthesis and adding their product to the product of the numbers in the j positions.

uv = 96.

To get the denominator, multiply the lengths of the vectors together. Then take the inverse cosine of the whole mess:

$cos^{-1}\theta=\frac{96}{111.64676}$ which returns an angle measure of 30.7. Plugging that all into the dot product formula:

$u*v=\sqrt{277}*\sqrt{45}cos(30.7)$ gives you a dot product of 96

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2 years ago