Let's go through the problem step-by-step to determine the normal force.
1. First, we know that Mary and Anne together apply a total force of [tex]\(48\)[/tex] newtons to just start moving the crate. This force corresponds to the force of static friction that they need to overcome.
2. The coefficient of static friction ([tex]\(\mu\)[/tex]) is given as [tex]\(0.11\)[/tex].
3. The equation that relates the force of static friction ([tex]\(f_{friction}\)[/tex]) to the normal force ([tex]\(N\)[/tex]) and the coefficient of static friction is:
[tex]\[
f_{friction} = \mu \times N
\][/tex]
4. Rearranging this equation to solve for the normal force ([tex]\(N\)[/tex]), we get:
[tex]\[
N = \frac{f_{friction}}{\mu}
\][/tex]
5. Substituting the given values into this equation:
[tex]\[
N = \frac{48 \text{ newtons}}{0.11}
\][/tex]
6. Calculating the above expression gives us:
[tex]\[
N \approx 436.36 \text{ newtons}
\][/tex]
Reviewing the answer choices:
A. [tex]\(2.0 \times 10^2\)[/tex] newtons = 200 newtons
B. [tex]\(2.6 \times 10^2\)[/tex] newtons = 260 newtons
C. [tex]\(3.9 \times 10^2\)[/tex] newtons = 390 newtons
D. [tex]\(4.0 \times 10^2\)[/tex] newtons = 400 newtons
E. [tex]\(4.3 \times 10^2\)[/tex] newtons = 430 newtons
The closest value to [tex]\(436.36\)[/tex] newtons is [tex]\(4.3 \times 10^2\)[/tex] newtons.
Therefore, the correct answer is:
E. [tex]\(4.3 \times 10^2\)[/tex] newtons
