Surface Area of a Cuboid: Calculate The Missing Side based on the formula

To solve the problem, let's recall the formula for calculating the surface area of a cube:

(width*length + height*width + height*length) *2

Let's substitute the known values into the formula, labelling the missing side X:

2*(3*7+7*X+3*X) = 122

2*(21+7x+3x) = 122

2(21+10x) = 122

Let's now expand the parentheses:

42+20x=122

Now we move terms:

20x=122-42

20x=80

Finally, simplify:

x=4

And that's the solution!

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information.
• Step 2: Apply the appropriate formula for the surface area of a cube.
• Step 3: Solve the equation to find the side length.

Now, let's work through each step:

Step 1: The problem gives us that the surface area of the cube is 24 cm².

Step 2: We'll use the formula for the surface area of a cube: $A = 6s^2$, where $A$ is the surface area and $s$ is the side length.

Step 3: Substitute the given surface area into the formula and solve for $s$:

$6s^2 = 24$

Divide both sides by 6 to isolate $s^2$:

$s^2 = \frac{24}{6} = 4$

Take the square root of both sides to solve for $s$:

$s = \sqrt{4} = 2$

Therefore, the solution to the problem is $s = 2$ cm.

To solve this problem, we'll follow these steps:

• Step 1: Identify the known information.
• Step 2: Apply the surface area formula for the cube.
• Step 3: Solve for the side length of the cube.

Now, let's work through each step:

Step 1: We know the total surface area of the cube is given as 24 cm².

Step 2: The formula for the surface area of a cube is:

$S = 6a^2$

where $S$ is the surface area and $a$ is the side length of the cube.

Step 3: We set the surface area equal to 24 cm² and solve for $a$:

$6a^2 = 24$

Divide both sides by 6:

$a^2 = 4$

Take the square root of both sides to solve for $a$:

$a = \sqrt{4} = 2 \text{ cm}$

Therefore, the length of each side of the cube is $2 \, \text{cm}$.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: The problem gives us a square face of the cuboid with side length 7 cm, meaning two dimensions are $7 \text{ cm}$ each. The total surface area of the cuboid is 406 cm².

Step 2: We'll use the formula for the surface area of a cuboid:
$\text{Surface area} = 2(lw + lh + wh)$

Let the dimensions be $l$ (length), $w = 7 \, \text{cm}$ (width), and $h = 7 \, \text{cm}$ (height).

The formula becomes:
$2(l \times 7 + l \times 7 + 7 \times 7) = 406$

Step 3: Simplify and solve for $l$.
Plug in the known values:
$2(7l + 7l + 49) = 406$

Simplify:
$2(14l + 49) = 406$

Divide by 2:
$14l + 49 = 203$

Subtract 49 from both sides:
$14l = 154$

Divide by 14:
$l = \frac{154}{14}$

So,
$l = 11 \, \text{cm}$

Therefore, the length of the cuboid is 11 cm.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information and relevant formula
• Step 2: Substitute the known values into the formula
• Step 3: Solve for the unknown variable (length)

Now, let's work through each step:

Step 1: Given the surface area of the cuboid is 124 cm², the width $w = 4$ cm, and the height $h = 2$ cm, we need to find the length $l$. The formula for the surface area of a cuboid is:

$2(lw + lh + wh) = \text{Surface Area}$

Step 2: Substitute the values into the equation:

$2(l \cdot 4 + l \cdot 2 + 4 \cdot 2) = 124$

Which simplifies to:

$2(4l + 2l + 8) = 124$

$2(6l + 8) = 124$

Step 3: Solve for $l$:

First, divide both sides by 2 to simplify:

$6l + 8 = 62$

Subtract 8 from both sides:

$6l = 54$

Divide by 6:

$l = 9$

Therefore, the length of the cuboid is $9$ cm.

The surface area formula for a cuboid is given by:

$SA = 2(lw + lh + wh)$

Substitute the given dimensions and surface area into this formula:

$752 = 2(12 \times 8 + 12 \times (X + 4) + 8 \times (X + 4))$

First, calculate each product:

• $12 \times 8 = 96$
• $12 \times (X + 4) = 12X + 48$
• $8 \times (X + 4) = 8X + 32$

Substitute these products back into the equation:

$752 = 2(96 + 12X + 48 + 8X + 32)$

Combine like terms inside the parentheses:

$752 = 2(176 + 20X)$

Distribute the 2:

$752 = 352 + 40X$

Isolate $X$ by subtracting 352 from both sides:

$400 = 40X$

Divide by 40:

$X = 10$

Thus, the value of $X$ is 10 cm.

Let's solve this problem step-by-step:

Step 1: Given the surface area $S = 486$, we know the formula for the surface area of a cube is:

• $S = 6a^2$

Step 2: We need to rearrange this formula to find $a$. The equation becomes:

• $a^2 = \frac{S}{6}$
• $a = \sqrt{\frac{S}{6}}$

Step 3: Substitute the given surface area into this equation:

$a = \sqrt{\frac{486}{6}}$

Step 4: Perform the division:

$a = \sqrt{81}$

Step 5: Calculate the square root:

$a = 9$

Now that we have found the side length, let's find the volume:

Step 6: Use the formula for the volume of a cube:

• $V = a^3$

Step 7: Substitute $a = 9$ into the volume formula:

$V = 9^3$

Step 8: Calculate the cube:

$V = 729$

Thus, the length of the side of the cube is $\mathbf{9}$ and the volume of the cube is $\mathbf{729}$.

The final answer matches the given multiple choice result:

$a=9,V=729$

To solve the problem, we utilize the surface area formula for a rectangular prism, where the given prism has a base of dimensions $x$ and $2x$, and an unknown height $h$. The full formula for surface area is given as:

$A = 2lw + 2lh + 2wh$

In this situation, $l = x$, $w = 2x$, and $h$ is our unknown. Substituting these values, the formula becomes:

$A = 2(x)(2x) + 2(x)h + 2(2x)h$

This simplifies to:

$A = 4x^2 + 2xh + 4xh$

Further simplification gives:

$A = 4x^2 + 6xh$

We are given the total surface area as $4x^2 + 24x$. Setting this equal to our expression:

$4x^2 + 6xh = 4x^2 + 24x$

Subtract $4x^2$ from both sides:

$6xh = 24x$

We can then divide both sides by $6x$ to solve for $h$:

$h = \frac{24x}{6x}$

This simplifies to:

$h = 4$

Thus, the height of the rectangular prism is $4$ cm.

To solve this problem, we'll follow these steps:

• Step 1: Use the formula for the surface area of a cuboid.
• Step 2: Substitute the dimensions and equate to the given surface area expression.
• Step 3: Solve algebraically for the missing dimension $y$.

Now, let's work through each step:

Step 1: Consider a cuboid with dimensions $l = 4$, $w = 3x$, and $h = 2y$.
The formula for the surface area is $SA = 2(lw + lh + wh)$.

Step 2: Substitute the dimensions into the formula:

$SA = 2(4 \times 3x + 4 \times 2y + 3x \times 2y)$

This simplifies to $SA = 2(12x + 8y + 6xy)$.

Further simplifying, we have $SA = 24x + 16y + 12xy$.

According to the problem, this is equal to $66x + 56$. Therefore, set:

$24x + 16y + 12xy = 66x + 56$

Step 3: Solve the equation:

Rearrange the terms:

$24x + 12xy + 16y = 66x + 56$

$12xy + 16y = 42x + 56$

Factor common terms:

$y(12x + 16) = 42x + 56$

Divide throughout by $(12x + 16)$:

$y = \frac{42x + 56}{12x + 16}$

To further simplify, note that both numerator and denominator can be reduced:

Factor out the greatest common divisor:

$y = \frac{14(3x + 4)}{4(3x + 4)}$

Cancel $(3x + 4)$:

$y = \frac{14}{4} = 3.5$

Therefore, the solution to the problem is $y = 3.5$ cm, matching the correct choice.

To solve this problem, we'll follow these steps:

• Step 1: Calculate the total painted surface area using the paint volume and coverage.
• Step 2: Set up the equation using the dimensions of the container and solve for width.

Now, let's work through each step:

Step 1: Calculate the total painted surface area
Given that Soledad uses $35\frac{1}{3}$ liters of paint, which is $\frac{106}{3}$ liters, and each liter covers $\frac{1}{3}$ square meters, the total painted surface area is:

$\text{Total Surface Area} = \left(\frac{106}{3}\right) \times 3 = 106 \text{ square meters}$

Step 2: Formulate the equation for the painted surface area
The surface area painted includes the two sides ($2(h \cdot l)$), two ends ($2(h \cdot w)$), and the top ($l \cdot w$) minus the bottom ($l \cdot w$).
The equation for the total surface area becomes:

$2(4 \cdot 12) + 2(4 \cdot w) + (12 \cdot w) = 106$

Simplifying the equation:

$96 + 8w + 12w = 106$

$96 + 20w = 106$

Solving for $w$:

$20w = 106 - 96$

$20w = 10$

$w = \frac{10}{20} = 0.5 \text{ meters}$

Therefore, the width of the container is $0.5 \text{ meters}$.