SAT · Math · Geometry and Trigonometry
Áreas y volúmenes de figuras y cuerpos, y cómo cambian al escalar.
Un cubo de lado 2 tiene volumen 8. Si el lado se duplica (4), el volumen es $64=8\times23$.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
A triangle has a base of length $14$ centimeters and a height of $9$ centimeters. What is the area, in square centimeters, of the triangle?
Respuesta numérica (escribes el número).
Respuesta: 63
The area of a triangle is $\frac{1}{2}bh=\frac{1}{2}(14)(9)=63$ square centimeters.
Forgetting the $\frac{1}{2}$ gives $126$.
In the figure shown, all angles are right angles. What is the area of the figure?
Respuesta correcta: C. $68$
The figure is a $10$-by-$8$ rectangle with a corner removed. The removed piece is $10-6=4$ units wide and $8-5=3$ units tall. So the area is $10\cdot 8-4\cdot 3=80-12=68$.
(Or split it: a $10$-by-$5$ rectangle plus a $6$-by-$3$ rectangle: $50+18=68$.) $80$ ignores the missing corner, $36$ is the perimeter, and $62$ uses $6$ by $2$ for the upper piece instead of $6$ by $3$.
A solid metal cube has a surface area of $150$ square centimeters. The cube is melted down, and all of the metal is used to make a solid right circular cylinder whose diameter is equal to the edge length of the cube. What is the height, in centimeters, of the cylinder?
Respuesta correcta: B. $\frac{20}{\pi}$
A cube has $6$ congruent square faces, so each face has area $\frac{150}{6}=25$ and the edge length is $5$ centimeters. The volume of metal is $53=125$ cubic centimeters. The cylinder’s diameter is $5$, so its radius is $\frac{5}{2}$, and $\pi\left(\frac{5}{2}\right)2h=125$. Then $\frac{25\pi}{4}h=125$, so $h=\frac{125\cdot 4}{25\pi}=\frac{20}{\pi}$.
$\frac{5}{\pi}$ uses the diameter $5$ as the radius. $\frac{24}{\pi}$ uses the surface area, $150$, as if it were the volume. $20$ leaves out $\pi$.