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2 years ago

Please answer all, thank you☺️

Mathematics

1 answer:

Reptile [31]2 years ago

3 0

Answer:

a) 50 degrees

b) 21 degrees

c) 64 degrees

d) 70 degrees

e) 105 degrees

Step-by-step explanation:

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Find the distance between a point (– 2, 3 – 4) and its image on the plane x+y+z=3 measured parallel to a line

Ber [7]

Answer:

The distance is:

$\displaystyle\frac{3\sqrt{142}}{10}$

Step-by-step explanation:

We re-write the equation of the line in the format:

$\displaystyle\frac{x+2}{3}=\frac{y+\frac{3}{2}}{2}=\frac{z+\frac{4}{3}}{\frac{5}{3}}$

Notice we divided the fraction of y by 2/2, and the fraction of z by 3/3.

In that equation, the director vector of the line is built with the denominators of the equation of the line, thus:

$\displaystyle\vec{v}=\left< 3, 2, \frac{5}{3}\right>$

Then the parametric equations of the line along that vector and passing through the point (-2, 3, -4) are:

$x=-2+3t\\y=3+2t\\\displaystyle z=-4+\frac{5}{3}t$

We plug them into the equation of the plane to get the intersection of that line and the plane, since that intersection is the image on the plane of the point (-2, 3, -4) parallel to the given line:

$\displaystyle x+y+z=3\to -2+3t+3+2t-4+\frac{5}{3}t=3$

Then we solve that equation for t, to get:

$\displaystyle \frac{20}{3}t-3=3\to t=\frac{9}{10}$

Then plugging that value of t into the parametric equations of the line we get the coordinates of the intersection:

$\displaystyle x=-2+3\left(\frac{9}{10}\right)=\frac{7}{10}\\\displaystyle y=3+2\left(\frac{9}{10}\right)=\frac{24}{5} \\\displaystyle z=-4+\frac{5}{3}\left(\frac{9}{10}\right)=-\frac{5}{2}$

Then to find the distance we just use the distance formula:

$\displaystyle d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}$

So we get:

$\displaystyle d=\sqrt{\left(-2-\frac{7}{10}\right)^2+\left(3-\frac{24}{5}\right)^2+\left(-4 +\frac{5}{2}\right)^2}=\frac{3\sqrt{142}}{10}$

3 0

3 years ago

Matthew made a fruit salad, in which the ratio of blueberries to red grapes is 8:6. If Matthew used 28 blueberries, how many red

Inga [223]

Answer:

Step-by-step explanation:

Let's look at some other ratios that equal 8:6 We could increase them or decrease them. For instance if we cut one in half we cut the other one in half, so 8:6 would be the same ratio as 4:3. With the same thinking if we double one we double the other so 8:6 s the same ratio as 16:12. Now, we want to go from 8 to 28. What do we multiply 8 by to get to it? Or, it might be easier to use 4:3 and see what you multiply 4 by. Then you multiply 3 by that same number to get the other number.

Let me know if that doesn't make sense.

4 0

3 years ago

The perimeter of a quadrant of a circle is 71.4cm. Find its area

Karolina [17]

Answer:

circle: 6490.9 cm², quadrant: 1622.7 cm²

Step-by-step explanation:

The circumference of the entire circle is known as 2πr and it is 4 times the given value, since a circle has 4 quadrants:

2πr = 4·71.4 = 285.6, so r = 285.6 / 2π ≈ 45.45

The area of a circle is πr², filling in the value for r we just found:

πr² = π 45.45² ≈ 6490.9 cm²