1.) The volume of a rectangular box is 64 ft3. If the width is 3 times longer than the height, and the length is 9 times longer than the height, find the dimensions of the box. Define your variables then show all work using the equation editor. Write a sentence that explains your answer.
2.)Steve and Tammy leave a campground, hiking on two different trails. Steve heads south and Tammy heads east. By lunchtime they are 8 mi apart. Tammy walked 4 miles more than Steve. Find the distance each person hiked. Use completing the square to find your answer rounded to the nearest tenth of a mile.
Just translate the English to Math:
let the height be x ft
"the width is 3 times longer than the height"
---> width = 3x
"the length is 9 times longer than the height"
-----> length = 9x
Now use the given to form your equation
x(3x)(9x) = 64
27x^3 = 64
take cube root of both sides:
3x = 4
x = 4/3
continue ...
2nd question, clearly a right-angled triangle, thus Pythagoras.
Steve's distance --- x
Tammy's distance -- x+4
x^2 + (x+4)^2 = 8^2
expand and solve the resulting quadratic
What "equation editor" are you talking about ?
i didn't mean to add that sorry
Equation editor is function within blackboard. It allows students to do step by step math problems with equations rather than using the ^2 functions like they would on a calculator. If you see that, all it means is they're probably in an online class or their homework is turned in through blackboard.
1.) Let's define our variables:
- Let "h" represent the height of the rectangular box.
- Since the width is 3 times longer than the height, we can represent it as "3h".
- Similarly, the length is 9 times longer than the height, so it can be represented as "9h".
The volume of a rectangular box is given by the formula: volume = length * width * height. In this case, the volume is 64 ft^3.
So, we have the equation: (9h) * (3h) * h = 64.
Let's simplify this equation using the equation editor:
\[27h^3 = 64.\]
Now, let's solve for "h" by finding the cube root of both sides:
\[h = \sqrt[3]{\frac{64}{27}}.\]
Calculating the cube root gives:
\[h = \frac{4}{3}.\]
Therefore, the height of the box is \(\frac{4}{3}\) ft.
To find the dimensions of the box, we can substitute this value of "h" back into our expressions for the width and length:
- Width = 3h = 3 * \(\frac{4}{3}\) = 4 ft.
- Length = 9h = 9 * \(\frac{4}{3}\) = 12 ft.
So, the dimensions of the box are: height = \(\frac{4}{3}\) ft, width = 4 ft, and length = 12 ft.
2.) Let's define our variables:
- Let "x" represent the distance Steve hiked.
- Since Tammy walked 4 miles more than Steve, we can represent her distance as "x + 4".
Using the Pythagorean Theorem, because Steve and Tammy walked perpendicular paths, we have the equation:
\[x^2 + (x+4)^2 = 8^2.\]
Now, let's simplify this equation using the equation editor:
\[x^2 + (x^2 + 8x + 16) = 64.\]
\[2x^2 + 8x - 48 = 0.\]
To solve this quadratic equation, we can use the completing the square method. We divide the equation by 2 to simplify it:
\[x^2 + 4x - 24 = 0.\]
Adding 24 to both sides to complete the square, we get:
\[x^2 + 4x = 24.\]
To complete the square, we take half of the coefficient of x (which is 4), square it (which is 4^2 = 16), and add it to both sides of the equation:
\[x^2 + 4x + 16 = 40.\]
Factoring the left side of the equation gives:
\[(x + 4)^2 = 40.\]
Taking the square root of both sides:
\[x + 4 = \sqrt{40}.\]
Taking the positive square root gives:
\[x + 4 = 6.32.\]
Subtracting 4 from both sides, we find:
\[x = 6.32 - 4 = 2.32.\]
Therefore, Steve hiked approximately 2.32 miles, and Tammy, who walked 4 miles more, hiked approximately 6.32 miles.
Please note that completing the square was used to find a decimal approximation rounded to the nearest tenth of a mile.