Please can someone help me with these last few questions on my homework? You don't have to give an answer because I would very much like to know how to do them?
An open box is formed by cutting squares with side lengths of 3 inches from each corner of a square piece of paper. What is a side length of the original paper if the box has a volume of 675 cubic inches?
21 inches
24 inches
33 inches
36 inches
The weekly cost (C) of growing and selling x acres of flowers is approximated by C = 0.2x2 12x + 240. How many acres of flowers would minimize these costs?
10 acres
20 acres
30 acres
60 acres
If the box originally had side s, then the new box has side s-6, since 3" was cut off each corner. The height of the box is just the 3" that was cut off. So,
s(s-6)^2 = 675
Now, just by inspection, 675 = 3*225 = 3*15^2
So, the box now has sides of 15, and height=3
The original paper was thus 21" on a side.
Sure! I'd be happy to help you with these questions and explain how to solve them.
For the first question, we are given that an open box is formed by cutting squares with side lengths of 3 inches from each corner of a square piece of paper. We need to find the side length of the original paper if the box has a volume of 675 cubic inches.
To solve this problem, we need to understand how the volume of a box is related to its dimensions. The volume of a box is given by the formula V = L × W × H, where L is the length, W is the width, and H is the height.
In this case, the height of the box is given as 3 inches (since we are cutting squares with side lengths of 3 inches from each corner). Let's denote the side length of the original paper as x inches.
The length and width of the base of the box can be calculated by subtracting twice the cut length (2 × 3 inches) from the side length of the original paper. So, the length and width would be (x - 2 × 3) inches.
Now, we have all the required dimensions to calculate the volume of the box: V = (x - 2 × 3) × (x - 2 × 3) × 3.
According to the problem, the volume of the box is 675 cubic inches. So, we can set up the equation:
675 = (x - 2 × 3) × (x - 2 × 3) × 3.
To solve this equation, you can simplify it and then solve for x. Once you find the value of x, you will have the side length of the original paper.
For the second question, we are given an equation to approximate the weekly cost (C) of growing and selling x acres of flowers: C = 0.2x^2 - 12x + 240. We need to find the number of acres of flowers that would minimize these costs.
To find the minimum cost, we need to find the vertex of the quadratic function represented by the given equation. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic equation in the form of ax^2 + bx + c.
In this case, the equation is C = 0.2x^2 - 12x + 240, so a = 0.2 and b = -12.
To find the value of x that minimizes the cost, substitute these values into the formula:
x = -(-12) / (2 * 0.2)
Simplifying this expression, we have:
x = 12 / 0.4
x = 30
Therefore, the number of acres of flowers that would minimize the costs is 30 acres.
I hope this helps you understand how to approach these problems! Let me know if you have any further questions.