Question

a rectangular piece of iron has sides with lengths of $7.08 \times 10^{-3}$ m, $2.18 \times 10^{-2}$ m, and $4.51 \times 10^{-3}$ m. what is the volume of the piece of iron?\
$\circ$ $6.96 \times 10^{-7}$ m$^3$\
$\circ$ $6.96 \times 10^7$ m$^3$\
$\circ$ $6.96 \times 10^{-18}$ m$^3$\
done

Explanation:

Step1: Recall the volume formula for a rectangular prism

The volume \( V \) of a rectangular prism (which a rectangular piece of iron is) is given by the product of its length \( l \), width \( w \), and height \( h \), so \( V = l \times w \times h \).

Step2: Substitute the given values

We have \( l = 7.08\times 10^{-3}\, \text{m} \), \( w = 2.18\times 10^{-2}\, \text{m} \), and \( h = 4.51\times 10^{-3}\, \text{m} \). So we need to calculate \( (7.08\times 10^{-3}) \times (2.18\times 10^{-2}) \times (4.51\times 10^{-3}) \).

Step3: Multiply the coefficients and the powers of 10 separately

First, multiply the coefficients: \( 7.08\times2.18\times4.51 \).

• \( 7.08\times2.18 = 15.4344 \)
• Then \( 15.4344\times4.51\approx 69.61 \approx 6.96\times 10^{1} \) (we round to match the significant figures in the answer choices)

Next, multiply the powers of 10 using the rule \( 10^{a}\times 10^{b}\times 10^{c}=10^{a + b + c} \).

• For the exponents: \( - 3+( - 2)+( - 3)=-8 \)? Wait, no, wait: Wait, \( 10^{-3}\times10^{-2}\times10^{-3}=10^{-3 - 2 - 3}=10^{-8} \)? Wait, no, wait, let's recalculate the exponents: \( - 3+(-2)+(-3)=-8 \)? Wait, no, the original numbers: \( 7.08\times 10^{-3} \), \( 2.18\times 10^{-2} \), \( 4.51\times 10^{-3} \). So exponents: \( - 3 + (-2)+(-3)=-8 \)? Wait, but the answer choices have exponents -7, 7, -18. Wait, maybe I made a mistake in calculation. Wait, let's recalculate the coefficient and exponents again.

Wait, \( 7.08\times2.18 = 15.4344 \); \( 15.4344\times4.51 \): Let's do \( 15\times4.51 = 67.65 \), \( 0.4344\times4.51\approx1.96 \), so total is \( 67.65 + 1.96 = 69.61 \), which is \( 6.96\times 10^{1} \). Now for the powers of 10: \( 10^{-3}\times10^{-2}\times10^{-3}=10^{-3-2 - 3}=10^{-8} \). Then multiply the coefficient and the power of 10: \( 6.96\times 10^{1}\times10^{-8}=6.96\times 10^{-7} \). Because \( 10^{1}\times10^{-8}=10^{1 - 8}=10^{-7} \).

Answer:

\( 6.96\times 10^{-7}\, \text{m}^3 \) (corresponding to the first option: \( 6.96\times 10^{-7}\, \text{m}^3 \))