Factoring to solve quadratic equations: Use the Zero-Product Property to solve each equation. Write your solution in roster form:
The volume of a storage tub shaped like a rectangular prism is 24 ft^3. The height of the tub is 3 feet. The width is w feet and the length is w+2ft. Use the formula V +lwh to find the value of w.
Please solve and explain. Thanks.
4, -2
To solve this problem, we will use the formula for the volume of a rectangular prism, V = lwh (length × width × height).
We are given that the volume of the tub is 24 ft³, the height is 3 ft, and the length is w + 2 ft. We need to find the value of w, which represents the width.
Substituting the given values into the formula, we have:
24 = (w + 2) × w × 3
To solve this equation, we will factor out the common factors on the right side of the equation:
24 = 3w(w + 2)
Now, we can apply the Zero-Product Property, which states that if a × b = 0, then either a = 0 or b = 0.
Setting each factor equal to zero and solving for w:
3w = 0
w = 0
w + 2 = 0
w = -2
Therefore, the possible values for w are 0 and -2.
However, in this case, we are considering the width of a tub, which cannot be negative. Therefore, we discard the value w = -2.
Thus, the only valid solution is:
w = 0
So, the width of the storage tub is 0 feet.
To solve the equation, we'll use the Zero-Product Property, which states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero.
The formula to calculate the volume of a rectangular prism is V = lwh, where V represents the volume, l represents the length, w represents the width, and h represents the height.
In this case, we have the following information from the problem:
- The volume of the storage tub (V) is 24 ft^3.
- The height of the tub (h) is 3 feet.
- The width of the tub (w) is unknown.
- The length of the tub (l) is w + 2 feet.
We can substitute these values into the volume formula and then solve for w:
24 ft^3 = (w + 2)ft * wft * 3ft
Now, let's simplify and rewrite the equation in quadratic form:
24 ft^3 = 3w^2 + 6w
To solve this quadratic equation, we need to rewrite it in standard form, where one side of the equation is zero:
3w^2 + 6w - 24 ft^3 = 0
Now, let's factor the equation:
3w^2 + 6w - 24 ft^3 = 0
3(w^2 + 2w - 8 ft^3) = 0
Now, we have a quadratic equation in factored form. To find the values of w, we set each factor equal to zero:
w^2 + 2w - 8 ft^3 = 0
(w + 4)(w - 2) = 0
Applying the Zero-Product Property, we get two possible values for w:
w + 4 = 0 or w - 2 = 0
Solving each equation separately, we find:
w + 4 = 0 => w = -4
w - 2 = 0 => w = 2
However, since we are dealing with lengths, we discard the negative value (-4) as length cannot be negative. So, the only solution is w = 2 feet.
Therefore, the value of w is 2 feet.
V = lwh
24 = 3w(w+2)
Multiply.
24 = 3w^2 + 6w
Put all terms on one side of the equation.
3w^2 + 6w -24 = 0
Factor.
(3w-12)(w+6) = 0
Solve for w and w+2.