Can you help me with this please can you show work please
1 answer:
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Answer:
Maximum volume of the box will be 7.41 cubic feet.
Step-by-step explanation:
Open box has been made from a metal sheet measuring 3 ft and 8 ft.
Let four square pieces were removed from the four corners with one side measurement x ft.
Volume of the open box = Length × width × height
Length of the box = (3 - 2x) ft
Width of the box = (8 - 2x) ft
Height of the box = x ft
Volume of the box = (3 - 2x)(8 - 2x)x
V = (24 - 6x - 16x + 4x²)x
V = 24x - 22x² + 4x³
Now we take the derivative of V with respect to x and equate the derivative to zero,
V' = 24 - 44x + 12x²
V' = 0
12x² - 44x + 24 = 0
3x² - 11x + 6 = 0
3x² - 9x - 2x + 6 = 0
3x(x - 3) - 2(x - 3) = 0
(3x - 2)(x - 3) = 0
(3x - 2) = 0
and (x - 3) = 0
Therefore, x = 3,
For x = 0.67
Length of the box = (3 - 2x) = 3 - 1.34
= 1.66 ft
Width of the box = (8 - 2x) = 8 - 1.34
= 6.66 ft
Volume of the box = 0.67 × 1.66 × 6.66
V = 7.41 cubic feet.
Similarly, for x = 3,
Length of the box = (3 - 2\times 3) = -3
which is negative but the length of the box can not be negative.
Therefore, maximum volume of the box will be 7.41 cubic feet.
Answer:
7. 183 in^2
8. 192.42 m^2
9. 239.4 ft^2
Step-by-step explanation:
7. The figure seems most easily seen as a trapezoid with a triangle removed. The trapezoid has bases of 26 and 14, and a height of 15. So, its area is ...
A = (1/2)(b1 +b2)(h) = (1/2)(26+14)(15) = 300 . . . . . . square inches
The triangle has a base of 26 and a height of 9, so an area of ...
A = (1/2)bh = (1/2)(26)(9) = 117 . . . . . . square inches
Then the shaded area is ...
300 in^2 -117 in^2 = 183 in^2
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8. The area of the donut can be found different ways. One way is to multiply the width of the shaded area (3.5 m) by the circumference of its centerline.
The centerline diameter is 21 m - 3.5 m = 17.5 m, so the circumference of that circle is ...
C = πd = π(17.5 m) = 17.5π m
Then the area of the shaded part is ...
(3.5 m)(17.5π m) = 61.25π m^2 ≈ 192.42 m^2
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9. As in problem 7, this can be treated as a parallelogram less a triangle. Here, we need to do a little work to find the area of the triangle.
The triangle of interest is a right triangle with hypotenuse 23.8 ft and one leg 21 ft. The Pythagorean theorem can be used to find the other leg:
h = √(23.8^2 -21^2) = √125.44 = 11.2
Then the area of the triangle is ...
A = (1/2)bh = (1/2)(21)(11.2) = 117.6 . . . ft^2
The area of the parallelogram (including the triangle) is ...
A = bh = (23.8 ft)(15 ft) = 357 ft^2
So, the shaded area is the difference ...
357 ft^2 -117.6 ft^2 = 239.4 ft^2
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<em>Comment on significant digits</em>
The area in problem 8 may be considered to be 192.325 m^2 if you are required to use 3.14 as the value for pi. Since that value is only good to 3 significant digits, it is quite pointless to report the answer to 6 significant digits when using that value. It is quite reasonable to use 192 m^2 for the answer, especially if you're required to round to whole numbers.
