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Varvara68 [4.7K]

3 years ago

9

∠A and \angle B∠B are vertical angles. If m\angle A=(3x-30)^{\circ}∠A=(3x−30) ∘ and m\angle B=(2x-9)^{\circ}∠B=(2x−9) ∘ , then f

ind the value of x.

1 answer:

jasenka [17]3 years ago

3 0

Answer:

x = 21

Step-by-step explanation:

Vertical angles are congruent, this

3x - 30 = 2x - 9 ( subtract 2x from both sides )

x - 30 = - 9 ( add 30 to both sides )

x = 21

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Write as an ordered pair. 4x+2y=16 and -4y+x=4

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Consider the right cone and right triangular prism below. Suppose that all measurements are labeled in centimeters.

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The prism has a volume about 340 cubic centimeters larger than the cone.

Step-by-step explanation:

<h3><u>Cone</u></h3>

<u />

<u>Formulas</u>

$\sf Surface\:area\:of\:a\:cone=\pi r \left(r+\sqrt{h^2+r^2}\right)$

$\textsf{Volume of a cone}=\sf \dfrac{1}{3} \pi r^2 h$

where:

r = radius of circular base
h = height perpendicular to the base

Given:

r = 3 cm
h = 6 cm

Substitute the given values into the formulas:

$\begin{aligned}\sf Surface\:area\:of\:cone & =\pi (3) \left(3+\sqrt{6^2+3^2}\right)\\ & = 3 \pi \left (3+\sqrt{36+9}\right)\\ & = 3\pi (3+\sqrt{45})\\ & = 3\pi(3+3\sqrt{5})\\ & = 91.5 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}$

$\begin{aligned}\textsf{Volume of cone} & =\dfrac{1}{3} \pi (3)^2 (6)\\& = \dfrac{54}{3} \pi \\ & = 18 \pi \\ & = 56.5\:\: \sf cm^3 \:(1 \: d.p.)\end{aligned}$

<h3><u>Prism</u></h3>

<u>Formulas</u>

<u /> $\textsf{Surface area of a prism}=\textsf{Total area of all the sides}$

$\textsf{Volume of a prism}=\sf \textsf{Area of base} \times height$

$\textsf{Area of a triangle}=\sf \dfrac{1}{2} \times base \times height$

$\textsf{Area of a rectangle}=\sf width \times length$

Given:

Height of triangular base = 10 cm
Base of triangular base = 8 cm
Height of prism = 10 cm

Find the <u>area of the triangular base</u> of the prism:

$\begin{aligned}\textsf{Area of the base} & = \dfrac{1}{2} \times 8 \times 10\\& = 40\:\: \sf cm^2\end{aligned}$

Find the third edge of the triangular base by using <u>Pythagoras Theorem</u>:

$\begin{aligned}a^2+b^2 & = c^2\\\implies 8^2+10^2 & = c^2\\164 & = c^2\\c & = \sqrt{164}\\c & = 2\sqrt{41}\end{aligned}$

Use the found values and the formulas to find the surface area of volume of the prism:

$\begin{aligned}\textsf{Surface area of prism} & = \sf 2\:triangles+3\:rectangles\\& = 2\left(40\right) + (10 \times 10)+(10 \times 8)+ (10 \times 2\sqrt{41})\\& = 80 + 100 + 80 + 20\sqrt{41}\\& = 388.1 \:\: \sf cm^2\:(1\:d.p.)\end{aligned}$

$\begin{aligned}\textsf{Volume of prism} & = 40 \times 10\\& = 400\:\:\sf cm^3 \:(1 \:d.p.)\end{aligned}$

<h3><u>Conclusion</u></h3>

The surface area and volume of the prism is <u>larger</u> than that of the cone.

<u>Difference between surface areas</u>:

388.1 - 91.5 = 296.6 ≈ 300 cm²

<u>Difference between volumes</u>:

400 - 56.5 = 343.5 ≈ 340 cm³

Therefore:

The prism has a surface area about 300 square centimeters larger than the cone.
The prism has a volume about 340 cubic centimeters larger than the cone.

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