Questions: Application of Quadratic Equations Question 5 of 11 (2 points) Question Attempt: 1 of Unlimited The height of a rectangular box is 8 ft. The length is 2 ft longer than thrice the width x. The volume is 680 ft^3. Part 1 of 4 (a) Write an equation in terms of x that represents the given relationship. The equation is 24x^2+16x-680=0. Part 2 of 4 (b) Solve the equation to find the dimensions of the given shape. The length is ft. Part 3 of 4 The width is ft.

Application of Quadratic Equations

Question 5 of 11 (2 points)  Question Attempt: 1 of Unlimited

The height of a rectangular box is 8 ft. The length is 2 ft longer than thrice the width x. The volume is 680 ft^3.

Part 1 of 4

(a) Write an equation in terms of x that represents the given relationship.

The equation is 24x^2+16x-680=0.

Part 2 of 4

(b) Solve the equation to find the dimensions of the given shape.

The length is ft.

Part 3 of 4

The width is ft.
Transcript text: 1.5 Application or Quadratic Equations Question 5 of 11 (2 points) | Question Attempt: 1 of Unlimited The height of a rectangular box is 8 ft. The length is 2 ft longer than thrice the width $x$. The volume is $680 \mathrm{ft}^{3}$. Part 1 of 4 (a) Write an equation in terms of $x$ that represents the given relationship. The equation is $24 x^{2}+16 x-680=0$. Part 2 of 4 (b) Solve the equation to find the dimensions of the given shape. The length is $\square$ ft. Part 3 of 4 The width is $\square$ ft.
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Solution

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Find the dimensions of the rectangular box given the volume and relationships between its dimensions.

Write the equation representing the relationship.

The height of the box is 8ft 8 \, \text{ft} , and the length L L is 2ft 2 \, \text{ft} longer than thrice the width x x . The volume V V is given by the equation:
V=LWH=(3x+2)x8=680 V = L \cdot W \cdot H = (3x + 2) \cdot x \cdot 8 = 680
This simplifies to:
24x2+16x680=0 24x^2 + 16x - 680 = 0

Factor the polynomial.

The polynomial 24x2+16x680 24x^2 + 16x - 680 can be factored as:
8(x5)(3x+17)=0 8 \left(x - 5\right) \left(3x + 17\right) = 0

Solve for x x .

Setting each factor to zero gives:
\[ x - 5 = 0 \quad \Rightarrow \quad x = 5
\]
3x+17=0x=173(not valid since width cannot be negative) 3x + 17 = 0 \quad \Rightarrow \quad x = -\frac{17}{3} \quad (\text{not valid since width cannot be negative})
Thus, the width is:
x=5ft \boxed{x = 5} \, \text{ft}

Calculate the length.

Using the width to find the length:
L=3x+2=3(5)+2=15+2=17ft L = 3x + 2 = 3(5) + 2 = 15 + 2 = 17 \, \text{ft}
Thus, the length is:
L=17ft \boxed{L = 17} \, \text{ft}

The width of the box is 5ft \boxed{5} \, \text{ft} and the length is 17ft \boxed{17} \, \text{ft} .

The width of the box is 5ft \boxed{5} \, \text{ft} .
The length of the box is 17ft \boxed{17} \, \text{ft} .

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