Question

1. challenge calculate the areas and densities. report the answers in the correct units.

a. the area of a rectangle with sides measuring 3×10^1 cm and 3×10^(-2) cm
b. the area of a rectangle with sides measuring 1×10^3 cm and 5×10^(-1) cm
c. the density of a substance having a mass of 9×10^5 g and a volume of 3×10^(-1) cm^3
d. the density of a substance having a mass of 4×10^(-3) g and a volume of 2×10^(-2) cm^3

Explanation:

Step1: Recall area formula for rectangle

The area formula for a rectangle is $A = l\times w$, where $l$ is the length and $w$ is the width. For density, the formula is $
ho=\frac{m}{V}$, where $m$ is mass and $V$ is volume.

Step2: Calculate area of rectangle in part a

$A_a=(3\times 10^{1})\times(3\times 10^{-2})$
Using the rule of exponents $a^m\times a^n=a^{m + n}$, we have $A_a=(3\times3)\times10^{1+( - 2)} = 9\times10^{-1}\text{ cm}^2$.

Step3: Calculate area of rectangle in part b

$A_b=(1\times 10^{3})\times(5\times 10^{-1})$
$A_b=(1\times5)\times10^{3+( - 1)}=5\times10^{2}\text{ cm}^2$.

Step4: Calculate density in part c

$
ho_c=\frac{9\times 10^{5}}{3\times 10^{-1}}$
Using the rule $\frac{a^m}{a^n}=a^{m - n}$, we get $
ho_c=\frac{9}{3}\times10^{5-( - 1)} = 3\times10^{6}\text{ g/cm}^3$.

Step5: Calculate density in part d

$
ho_d=\frac{4\times 10^{-3}}{2\times 10^{-2}}$
$
ho_d=\frac{4}{2}\times10^{-3-( - 2)}=2\times10^{-1}\text{ g/cm}^3$.

Answer:

a. $9\times 10^{-1}\text{ cm}^2$
b. $5\times 10^{2}\text{ cm}^2$
c. $3\times 10^{6}\text{ g/cm}^3$
d. $2\times 10^{-1}\text{ g/cm}^3$