Find the lateral area, the total area and the volume of a circular cone that is circumscribed to a sphere of radius $\,r\,$ and whose axial cut is an equilateral triangle.
In the following picture: $\,\overline{PP'}\,$ is the diameter of the sphere whose center is $\,O\,$,$\;M\,$ é is the center of a intersection with a plane perpendicular to $\,\overline{PP'}\,$. Also the measures are $\,\overline{AP}\,=\,6\,cm\;$ and $\,\overline{AP'}\,=\,8\,cm\;$. Calculate the area of the cicle whose center is $\,M\,$.
answer: The area is $\,\frac{\,576\,\pi\;}{\;25\;}\;cm^2$ ×
In the picture the line r is parallel to the line s . Find the value of $\,\hat{\,\alpha\,}\,$.
The expression $\phantom{X}[\dfrac{\sqrt{a\;+\;b}\;-\;\sqrt{a}}{b}]^{-1}\phantom{X}$ , where $\;a\;$ and $\;b\;$ are positive numbers, is equivalent to:
What is the vertical height (altitude) of the right cone whose radius of the circular base is equal to $\;\sqrt{3}\,$ centimeters and the slant height is 5 centimeters?
answer:
Thinking:
The slant height of the cone is the distance from the apex to a point in the perimeter of the base. If the cone is a right cone, the length of all the slant heights are the same.
Solution:
$\,\left.\begin{array}{rcr} \mbox{slant height }\phantom{XXXX}\;\,\rightarrow\, & \;S_h\mbox{ = 5 cm }\; \\ \,\mbox{radius of the base}\phantom{XX} \rightarrow\, & R\,=\,\sqrt{3}\\ \mbox{Pythagorean theorem}\, \rightarrow\, & (S_h)^{\large 2}\,=\,H^{\large 2}\,+\,R^{\large 2}\; \\ \end{array} \right\}\;\Rightarrow\;$
the vertical height of the cone is $\,H\,=\,\sqrt{22}\,$ cm ×
A machine working 6 hours each day produces 3000 hair clips in 5 days. How many hair clips will be produced if the machine is working 4 hours each day during 8 days?
A small spur gear with 30 teeth drives another gear with 25 teeth. When the first gear has made 450 revolutions, how many revolutions has the second one made?
Three points are given in the (x,y)-plane: $\;A=(1,2)\;$, $\;B=(2,-2)\;$ and $\;C=(4,3)\;$. The equation of the line passing through the point $\;A\;$ and the midpoint of the segment $\;\overline{BC}\;$ is:
The length of the two parallel sides of a right trapezoid are 6 cm and 8 cm, respectively top and bottom bases, and the altitude is 4 cm. The distance between the intersection of the two straight lines that contain the non-parallel legs and the midpoint of the major base is:
The lantern shown in the picture is suspended by two strings perpendicular to each other. Knowing the length of each string is 1/2 and 6/5, respectively, what is the distance between the lantern and the ceiling?
In the picture below, ABCD is a rectangle, $\,\overline{AB}\,=\,4\,$, $\,\overline{BC}\,=\,1\;$ and $\,\overline{DE}\,=\,\overline{EF}\,=\,\overline{FC}\;$. The length of $\,\overline{BG}\,$ is:
In the picture below, ABFG and BCDE are squares and their each one sides measures, respectively, a and b. Knowing that $\;\overline{AG}\,=\,\overline{CD}\,+\,2\;\,$ and the perimeter of the triangle ACG is 12, we can assure that a e b are both inside the interval:
Given that: $\,\overline{MP}\;\bot\;s\,$;$\;\overline{MQ}\;\bot\;t\,$;$\;\overline{MQ}\;\bot\;\overline{PQ}\,$;$\;\overline{MP}\,=\,6$ The length of $\,\overline{PQ}\,$ é is:
The picture shows the rhombus ABCD and the point A is the center of the circle that has a 4 cm length radius. Find the area of the rhombus in square centimeters.
(FESP - 1991) An equilateral triangle ABC is inscriibed in a circle whose radius is 6 cm length. The triangle is intercepted by a diameter MN of the circle, forming a trapezoid, as shown in the picture below. We can say that the quotient of the triangle ABC area divided by the trapezoid APQC area is:
AB is the diameter of the circle whose center is O and the triangle ABC is inscribed in. The quotient $\,\dfrac{s}{S}\,$ where the area $\,s\,$ of the triangle ACO is divided by the area $\,S\,$ of the triangle COB is:
The circle as shown in the picture below, with center P and radius 2, is tangent to three sides of the rectangle ABCD. Given that the total area of the rectangle is 32, find the distance between the point P and the diagonal AC.
The circles in the picture are externally tangent. The distance between the centers $\,\overline{OC}\,$ is 28 cm and the difference between their radii is 8 cm. Find the length of each radius.
The length of the arc $\,\stackrel \frown{AB}\,$, as in the picture, is 22 cm and O is the center of the circle. The perimeter of the circle (circumference) is:
(MACKENZIE - 1969) Knowing the set $\,\mathbb{A}\,=\,\lbrace\,\lbrace\,1\,\rbrace , \,\lbrace\,2\,\rbrace,\,\lbrace\,1,\,2\,\rbrace\,\rbrace\,\;$, which one of the following statements is correct?
