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 right circular cone has a volume of $50$ cubic centimeters. A second right circular cone has a radius that is $2$ times the radius of the first cone and a height that is $\frac{1}{2}$ the height of the first cone. What is the volume, in cubic centimeters, of the second cone?
Respuesta numérica (escribes el número).
Respuesta: 100
The volume of a cone is $V=\frac{1}{3}\pi r2h$. Doubling the radius multiplies $r2$ by $4$, and halving the height multiplies $h$ by $\frac{1}{2}$. The volume is multiplied by $4\cdot\frac{1}{2}=2$: $2\cdot 50=100$ cubic centimeters.
A common mistake is thinking the two changes cancel ($50$), forgetting that the radius is squared.