14
Active contributors today

## Geometry help? Volume of a cone.

Daniel H.
Featured 3 weeks ago

The volume of a cone is $V = \frac{\setminus \pi {r}^{2} h}{3}$

#### Explanation:

$1014 \setminus \pi = \frac{\setminus \pi {r}^{2} \cdot 18}{3}$

The $\setminus \pi$ on each side of the equal sign will cancel, so

$1014 = \frac{{r}^{2} \cdot 18}{3}$

Multiply both sides by 3

$3042 = {r}^{2} \cdot 18$

Then divide both sides by 18

$169 = {r}^{2}$

Then, take the square root of both sides

$\sqrt{169} = {\sqrt{r}}^{2}$

$\pm 13 = r$

Since this is a distance , use the positive square root since distances can't be negative, so r = 13.

Then, the circumference of a circle is $2 \setminus \pi r$

So, $2 \cdot 13 \setminus \pi \to 26 \setminus \pi$

That is your answer and is an exact value since it is in terms of $\setminus \pi$

## Two opposite sides of a parallelogram have lengths of #3 #. If one corner of the parallelogram has an angle of #pi/4 # and the parallelogram's area is #36 #, how long are the other two sides?

sankarankalyanam
Featured 3 weeks ago

Other two parallel sides are #color(green)(12sqrt2# long each.

#### Explanation:

Area of parallelogram ${A}_{p} = b h = b a \sin \theta$ as $h = a \sin \theta$

Given : $b = 3 , {A}_{p} = 36 , \theta = \frac{\pi}{4}$

To find a.

$a = {A}_{p} / \left(b \sin \theta\right) = {\cancel{36}}^{\textcolor{b r o w n}{12}} / \left(\cancel{3} \sin \left(\frac{\pi}{4}\right)\right) = \frac{12}{\frac{1}{\sqrt{2}}} = 12 \sqrt{2}$

## Find the equation of the circle with center (3, 2) and (9, 3) a point on the circle. What is the equation?

smendyka
Featured 3 weeks ago

See a solution process below:

#### Explanation:

If we find the distance between the center of the circle and the point on the circle we will know the radius of the circle.

The formula for calculating the distance between two points is:

$d = \sqrt{{\left(\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}\right)}^{2} + {\left(\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}\right)}^{2}}$

Substituting the values from the points in the problem gives:

$d = \sqrt{{\left(\textcolor{red}{9} - \textcolor{b l u e}{3}\right)}^{2} + {\left(\textcolor{red}{3} - \textcolor{b l u e}{2}\right)}^{2}}$

$d = \sqrt{{6}^{2} + {1}^{2}}$

$d = \sqrt{36 + 1}$

$d = \sqrt{37}$

Therefore the radius of the circle is: $\sqrt{37}$

The equation for a circle is:

${\left(x - \textcolor{red}{a}\right)}^{2} + {\left(y - \textcolor{red}{b}\right)}^{2} = {\textcolor{b l u e}{r}}^{2}$

Where $\left(\textcolor{red}{a} , \textcolor{red}{b}\right)$ is the center of the circle $\left(\textcolor{red}{3} , \textcolor{red}{2}\right)$ for this problem.

And $\textcolor{b l u e}{r}$ is the radius of the circle. $\textcolor{b l u e}{\sqrt{37}}$ for this problem.

Substituting these values gives:

${\left(x - \textcolor{red}{3}\right)}^{2} + {\left(y - \textcolor{red}{2}\right)}^{2} = {\textcolor{b l u e}{\left(\sqrt{37}\right)}}^{2}$

${\left(x - \textcolor{red}{3}\right)}^{2} + {\left(y - \textcolor{red}{2}\right)}^{2} = \textcolor{b l u e}{37}$

## One side of a rectangle is 10 cm longer than its width.The area of rectangle is 96 sq.cm. Find the dimensions of the rectangle?

Mahek :)
Featured 2 weeks ago

#color (red)(=6cm & 16cm#

#### Explanation:

Let one side of the rectangle be $x$, Other side=$x + 10$

Area of rectangle= $l \times b$

$96 = x \times \left(x + 10\right)$

$96 = {x}^{2} + 10 x$

${x}^{2} + 10 x - 96 = 0$

Identity:
#color(magenta)(x^2+(a+b)x+ab=(x+a)(x+b)#

${x}^{2} + 16 x - 6 x - 96 = 0$

$x \left(x - 6\right) + 16 \left(x - 6\right) = 0$

$\left(x - 6\right) \left(x + 16\right) = 0$

$\therefore x - 6 = 0 \mathmr{and} x + 16 = 0$

$I f x - 6 + 0 ,$ then $x = 6$

$I f x + 16 = 0 ,$ then $x = - 16$

Since $x$ cannot take a $- v e$ value, x=6

$\therefore ,$ the sides are #=x & x+10#

#color (red)(=6cm & 16cm#

~Hope this helps! :)

## If the largest angle of an isosceles triangle measures 124 degrees what is the measure of the smallest angle?

Douglas K.
Featured 2 weeks ago

$\angle A = \angle B = {28}^{\circ}$

#### Explanation:

If one angle of an isosceles triangle is greater than ${60}^{\circ}$, then the other two angles must be equal. You can use this fact and the fact that the sum of interior angles is ${180}^{\circ}$ to find the measure of those angles.

Given an isosceles triangle with $\angle C = {124}^{\circ}$, then we know know that:

$\angle A = \angle B$

And

$\angle A + \angle B + \angle C = {180}^{\circ}$

Substitute $\angle A = \angle B$:

$\angle A + \angle A + \angle C = {180}^{\circ}$

Substitute $\angle C = {124}^{\circ}$:

$2 \angle A + {124}^{\circ} = {180}^{\circ}$

$2 \angle A = {56}^{\circ}$

$\angle A = \angle B = {28}^{\circ}$

## Shown below is a sphere, cone and cube. The surface area of the sphere is equal to the sum the surface areas of the cone and cube. so what is Y's number?

Somebody N.
Featured 6 days ago

$\textcolor{b l u e}{y = r = \sqrt{9 + \frac{96}{\pi}} \approx 6.2895}$

#### Explanation:

First calculate the surface areas of the cone and cube.

The surface area of the cube:

We have 6 faces each having a surface area of:

$8 \cdot 8 = 64 {\text{cm}}^{2}$

Total surface area:

$6 \cdot 64 = 384 {\text{cm}}^{2}$

The surface area of a cone is given by:

$\boldsymbol{A = \pi r \left(r + \sqrt{{h}^{2} + {r}^{2}}\right)}$

$h$ is the height of the cone. We are given the length of the slope, so we will need to find $h$.

We can do this by using Pythagoras's theorem.

If the slope is the hypotenuse, and the radius is one of the sides of a right triangle, then the missing side $h$ is:

$h = \sqrt{{\left(9\right)}^{2} - {\left(3\right)}^{2}} = \sqrt{72} = 6 \sqrt{2}$

Putting known values into the formula:

$A = \pi \left(3\right) \left(\left(3\right) + \sqrt{{\left(6 \sqrt{2}\right)}^{2} + {\left(3\right)}^{2}}\right)$

$A = \pi \left(3\right) \left(\left(3\right) + \sqrt{\left(72 + 9\right)}\right)$

$A = \pi \left(3\right) \left(12\right) = 36 \pi$

Surface area of sphere is the sum of the surface area of the cube and the cone:

$\left(36 \pi + 384\right) {\text{cm}}^{2}$

Surface area of a sphere is given by:

$\boldsymbol{A = 4 \pi {r}^{2}}$

$\therefore$

$4 \pi {r}^{2} = 36 \pi + 384$

We need to find $r$ which will be our $y$ value:

$4 \pi {r}^{2} = 36 \pi + 384$

Divide by $4 \pi$:

${r}^{2} = \frac{36 \pi}{4 \pi} + \frac{384}{4 \pi}$

${r}^{2} = 9 + \frac{96}{\pi}$

Taking square root:

$r = \sqrt{9 + \frac{96}{\pi}}$

You can leave it in this form for an exact answer, or its approximate value of:

$\textcolor{b l u e}{y = r = \sqrt{9 + \frac{96}{\pi}} \approx 6.2895}$

##### Questions
• · 50 minutes ago
• · An hour ago
• · 2 hours ago
• · 2 hours ago
• · 3 hours ago
• · 4 hours ago
• · 6 hours ago
• 7 hours ago
• · 7 hours ago
• · 8 hours ago
• · 11 hours ago