The area of a playing card is 60 square centimeters. The perimeter is 32 centimeters. What are the dimensions of the playing card?
we’re looking for values of xy=60 (area) and 2x+2y = 32 (perimeter).
So we rearrange the 2nd equation to 2y = 32-2x.
Then y = 16-x. Put this equation into the first one to get x(16-x) =60, then expand to get 16x - x^2 -60 = 0.
Then put this into the quadratic equation to get x1= 6 and x2=10. Then plug these values into y = 16-x.
10=16-6, 6=16-10.
So the y values are 10 and 6.
As they come in pairs, x1=6, y1=10. And x1=10, y1=6.
As playing cards are usually taller than they are wide, the dimensions are in 10cm (height) x 6cm (width)
To find the dimensions of the playing card, we need to use the formulas for the area and perimeter of a rectangle.
Let's assume the length of the playing card is L and the width is W.
The area of a rectangle is given by the formula:
Area = Length x Width
In this case, the area of the playing card is given as 60 square centimeters, so we have:
60 = L x W ---(Equation 1)
The perimeter of a rectangle is given by the formula:
Perimeter = 2 x (Length + Width)
In this case, the perimeter of the playing card is given as 32 centimeters, so we have:
32 = 2(L + W)
Divide each side of the equation by 2 to simplify:
16 = L + W ---(Equation 2)
We now have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of L and W.
To solve the equations, we can use substitution or elimination method. Let's solve by substitution method.
From Equation 2, we can express L in terms of W:
L = 16 - W
Substitute this value of L in Equation 1:
60 = (16 - W) x W
Expand the equation:
60 = 16W - W^2
Rearrange the equation:
W^2 - 16W + 60 = 0
Now, we can solve this quadratic equation either by factoring or using the quadratic formula.
Factoring the equation:
(W - 6)(W - 10) = 0
Setting each factor equal to zero, we get:
W - 6 = 0 or W - 10 = 0
Solving each equation for W gives us two possible values for the width:
W = 6 or W = 10
Substituting these values back into Equation 2, we can find the corresponding lengths.
For W = 6:
L = 16 - 6
L = 10
For W = 10:
L = 16 - 10
L = 6
Therefore, the dimensions of the playing card can be:
Length = 10 cm, Width = 6 cm
or
Length = 6 cm, Width = 10 cm
To find the dimensions of the playing card, we can set up a system of equations using the given information.
Let's assume the length of the card is 'L' centimeters and the width is 'W' centimeters.
We are given two pieces of information:
1) The area of the card is 60 square centimeters, so we have the equation:
L × W = 60
2) The perimeter of the card is 32 centimeters, so we have the equation:
2L + 2W = 32
Now, we can solve this system of equations to find the dimensions.
1) Rearrange the first equation to find W in terms of L:
W = 60 ÷ L
2) Substitute this expression for W into the second equation:
2L + 2(60 ÷ L) = 32
3) Simplify the equation:
2L + 120 ÷ L = 32
4) Multiply through by L to eliminate the denominator:
2L^2 + 120 = 32L
5) Rearrange the equation:
2L^2 - 32L + 120 = 0
6) Divide through by 2 to simplify:
L^2 - 16L + 60 = 0
Now, we can use factoring or the quadratic formula to solve this equation. In this case, the equation factors as:
(L - 10)(L - 6) = 0
So, we have two possible solutions:
a) L - 10 = 0 --> L = 10
b) L - 6 = 0 --> L = 6
If we substitute these values of L back into our equation for W, we can find the corresponding dimensions for each value of L:
For L = 10:
W = 60 ÷ 10 = 6
For L = 6:
W = 60 ÷ 6 = 10
Therefore, the two possible sets of dimensions for the playing card are:
1) Length = 10 cm, Width = 6 cm
2) Length = 6 cm, Width = 10 cm