Answer:
7) 29 bottles; $3.19 each; 8 hours 24 minutes
8) 12x^2 +40x +28; 4x^2 +24x +36; d = 4√2(x +2)
Step-by-step explanation:
7)
a) 275 mL = 0.275 L, so the number of small bottles that can be filled is ...
8/0.275 ≈ 29.1
29 fill bottles can be filled
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b) In order make 175% profit, the revenue from the sale of the 29 bottles must be ...
$33.65 × 275% = $92.54
Then each bottle must be priced at ...
$92.54/29 = $3.19
The price for each bottle must be $3.19.
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c) The number of staff-hours required for 45 containers is ...
(4)(5 1/4) = 21 . . . . staff-hours
Then the work rate per container is ...
(21 staff·hours)/(45 containers) = 7/15 staff·hours/container
= 28 staff·minutes/container
Then for 3 staff and 54 containers, the time required is ...
(28 staff·minutes/container)(54 containers)/(3 staff) = 504 minutes
= 8 hours, 24 minutes
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8)
a) The area is the product of length and width. The 4 rectangles will have an area 4 times the area of one rectangle.
A = 4(3x +7)(x +1) = 4(3x^2 +10x +7) = 12x^2 +40x +28
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b) The edge dimension of the shaded square is ...
(3x +7) -(x +1) = 2x +6
The area of the shaded square is the square of this length, so is ...
A = (2x +6)^2 = 4x^2 +24x +36
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c) The edge length of the larger square is the sum of the length and width of the surrounding rectangles:
s = (3x +7) +(x +1) = 4x +8 = 4(x +2)
The diagonal length is √2 times this edge length*, so is ...
d = s√2
d = 4√2(x +2)
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* If you're not allowed to make use of this fact without deriving it, consider the hypotenuse of an isosceles right triangle with side lengths 1. By the Pythagorean theorem, the hypotenuse (c) has length ...
c^2 = 1^2 +1^2
c = √2 . . . . . . . . . take the square root
This ratio of sides and hypotenuse (1:1:√2) applies to any isosceles right triangle, so applies to the diagonal of the square in this problem.
