Certainly! Here is a detailed step-by-step solution for both questions:
1. Cube Dimensions with Surface Area 294 cm²:
We know that the surface area (SA) of a cube can be given by the formula:
[tex]\[ SA = 6a^2 \][/tex]
where [tex]\( a \)[/tex] is the length of one side of the cube.
Step-by-Step Solution:
1. Given the surface area [tex]\( SA = 294 \)[/tex] cm²:
[tex]\[ 6a^2 = 294 \][/tex]
2. Divide both sides of the equation by 6:
[tex]\[ a^2 = \frac{294}{6} \][/tex]
[tex]\[ a^2 = 49 \][/tex]
3. Solve for [tex]\( a \)[/tex] by taking the square root of both sides:
[tex]\[ a = \sqrt{49} \][/tex]
[tex]\[ a = 7 \][/tex]
So, the length of each side of the cube is:
[tex]\[ \boxed{7 \text{ cm}} \][/tex]
2. Length of the Other Parallel Side of a Trapezoid:
We know the formula for the area (A) of a trapezoid is:
[tex]\[ A = \frac{1}{2} \times (a + b) \times h \][/tex]
where:
- [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two parallel sides,
- [tex]\( h \)[/tex] is the height (distance between the parallel sides),
- A is the area.
Given:
- [tex]\( a = 5.5 \)[/tex] meters,
- [tex]\( h = 3.4 \)[/tex] meters,
- [tex]\( A = 22.61 \)[/tex] m².
Step-by-Step Solution:
1. Substitute the known values into the area formula:
[tex]\[ 22.61 = \frac{1}{2} \times (5.5 + b) \times 3.4 \][/tex]
2. Multiply both sides by 2 to clear the fraction:
[tex]\[ 45.22 = (5.5 + b) \times 3.4 \][/tex]
3. Divide both sides by 3.4:
[tex]\[ \frac{45.22}{3.4} = 5.5 + b \][/tex]
[tex]\[ 13.3 = 5.5 + b \][/tex]
4. Solve for [tex]\( b \)[/tex] by subtracting 5.5 from both sides:
[tex]\[ b = 13.3 - 5.5 \][/tex]
[tex]\[ b = 7.8 \][/tex]
So, the length of the other parallel side is:
[tex]\[ \boxed{7.8 \text{ meters}} \][/tex]
These are the step-by-step solutions showing all work.
