Answer :
To solve the problem, let's start by setting up our equations based on the information provided.
1. Define Variables:
- Let [tex]\( w \)[/tex] be the width of the rectangle.
- According to the problem, the length [tex]\( l \)[/tex] of the rectangle is 7.8 cm more than 4 times the width. So, we can express the length as:
[tex]\[
l = 4w + 7.8
\][/tex]
2. Perimeter Formula:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[
P = 2(l + w)
\][/tex]
- We are given the perimeter is 94.6 cm, so:
[tex]\[
2(l + w) = 94.6
\][/tex]
3. Substitute the Length Expression:
- Substitute the expression for [tex]\( l \)[/tex] from step 1 into the perimeter equation:
[tex]\[
2((4w + 7.8) + w) = 94.6
\][/tex]
4. Simplify the Equation:
- Combine the terms inside the parentheses:
[tex]\[
2(5w + 7.8) = 94.6
\][/tex]
5. Expand and Solve for [tex]\( w \)[/tex]:
- Expand the equation:
[tex]\[
10w + 15.6 = 94.6
\][/tex]
- Solve for [tex]\( w \)[/tex] by isolating it on one side of the equation:
[tex]\[
10w = 94.6 - 15.6
\][/tex]
[tex]\[
10w = 79
\][/tex]
[tex]\[
w = \frac{79}{10} = 7.9
\][/tex]
6. Find the Length [tex]\( l \)[/tex]:
- Use the width to find the length:
[tex]\[
l = 4w + 7.8 = 4(7.9) + 7.8
\][/tex]
[tex]\[
l = 31.6 + 7.8 = 39.4
\][/tex]
The dimensions of the rectangle are:
- Width = 7.9 cm
- Length = 39.4 cm
Therefore, the correct option is:
d. length [tex]\( = 39.4 \)[/tex] cm; width [tex]\( = 7.9 \)[/tex] cm.
1. Define Variables:
- Let [tex]\( w \)[/tex] be the width of the rectangle.
- According to the problem, the length [tex]\( l \)[/tex] of the rectangle is 7.8 cm more than 4 times the width. So, we can express the length as:
[tex]\[
l = 4w + 7.8
\][/tex]
2. Perimeter Formula:
- The perimeter [tex]\( P \)[/tex] of a rectangle is given by:
[tex]\[
P = 2(l + w)
\][/tex]
- We are given the perimeter is 94.6 cm, so:
[tex]\[
2(l + w) = 94.6
\][/tex]
3. Substitute the Length Expression:
- Substitute the expression for [tex]\( l \)[/tex] from step 1 into the perimeter equation:
[tex]\[
2((4w + 7.8) + w) = 94.6
\][/tex]
4. Simplify the Equation:
- Combine the terms inside the parentheses:
[tex]\[
2(5w + 7.8) = 94.6
\][/tex]
5. Expand and Solve for [tex]\( w \)[/tex]:
- Expand the equation:
[tex]\[
10w + 15.6 = 94.6
\][/tex]
- Solve for [tex]\( w \)[/tex] by isolating it on one side of the equation:
[tex]\[
10w = 94.6 - 15.6
\][/tex]
[tex]\[
10w = 79
\][/tex]
[tex]\[
w = \frac{79}{10} = 7.9
\][/tex]
6. Find the Length [tex]\( l \)[/tex]:
- Use the width to find the length:
[tex]\[
l = 4w + 7.8 = 4(7.9) + 7.8
\][/tex]
[tex]\[
l = 31.6 + 7.8 = 39.4
\][/tex]
The dimensions of the rectangle are:
- Width = 7.9 cm
- Length = 39.4 cm
Therefore, the correct option is:
d. length [tex]\( = 39.4 \)[/tex] cm; width [tex]\( = 7.9 \)[/tex] cm.
