想問3條中3數學題

1. If the sum of threeconsecutive numbers is less than 49,what is the largest possible value of the smallestnumber?

The three numbers are integers.
Let x be the smallest number.
Then, the other two numbers are (x + 1) and (x + 2).

x + (x + 1) + (x + 2) < 49
3x + 3 < 49
3x < 46
x < 15 + (1/3)

Hence, the largest possible value of the smallestnumber 15.


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2. Ben has 20 coins whichconsist only of $2-coins and $5-coins.If the total amount of these coins is notless than $85,how many $2-coins can he have at most?

Let n be the number of $2-coins.
Then, the number of $5-coins = 20 - n

The amount:
2n + 5(20 - n) ≥ 85
2n + 100 - 5n ≥ 85
-3n ≥ -15
3n ≤ 15
n ≤ 5

Hence, he can have at most 5 $2-coins.


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3. The length of a rectangleis equal to 3 times the width of the rectangle minus 8 cm. If the perimeter of therectangle is not less than 24 cm, find the minimum possiblearea of the rectangle.

Let y cm be the width of the rectangle.
Then, the length of the rectangle = (3y - 8) cm

perimeter:
2y + 2(3y - 8) ≥ 24
2y + 6y - 16 ≥ 24
8y ≥ 40
y ≥ 5
3y - 8 ≥ 7

Area of the rectangle
= y(3y - 8) cm²
≥ 5 * 7 cm²
= 35 cm²

Hence, the minimum possible area of the rectangle is 35cm².