a rectangle is 7 ft by 9ft. a second is similar to the first with a perimeter of 260 ft. what are the dimensions of the second rectangle?
if the new one is similar to the old one, then it must be
7x by 9x
and its perimeter would be 2(7x+9x) or 32x
so 32x = 260
x = 8.125
so the new one is 7(8.125) by 9(8.125)
or
56.875 by 73.125
check : 73.125/56.875 = 73125/56875 = 2925/2275 = 9/7 --- ratio works
perimeter = 2(56.875) + 2(73.125) = 260 --- checks!
To find the dimensions of the second rectangle, we can use the concept of similarity. Similar figures have corresponding angles that are equal and corresponding sides that are proportional.
Given that the first rectangle is 7 ft by 9 ft, we can find its perimeter by adding the lengths of all four sides, which is equal to 2 times the length plus 2 times the width.
Perimeter of the first rectangle = 2(Length + Width) = 2(7 ft + 9 ft) = 2(16 ft) = 32 ft.
Now, we are given that the second rectangle is similar to the first one but has a perimeter of 260 ft. Since the second rectangle is similar to the first one, the ratios of corresponding sides will be equal.
Let's assume the length of the second rectangle is 'x' ft. Therefore, the width of the second rectangle will be 'y' ft.
We can set up a proportion to find the dimensions of the second rectangle:
\( \frac{x}{y} = \frac{7}{9} \)
Now, we can calculate the perimeter of the second rectangle:
Perimeter of the second rectangle = 2(x + y) = 260 ft.
Substituting the given ratio, we can solve for 'x' and 'y':
\( \frac{x}{y} = \frac{7}{9} \)
\( 2(x + y) = 260 \)
To simplify the calculation, we can cross-multiply the first proportion:
\( 9x = 7y \) [equation 1]
Next, we can solve for 'x' in terms of 'y' by rearranging equation 1:
\( x = \frac{7y}{9} \)
Now, substitute the value of 'x' in the equation for the perimeter of the second rectangle:
\( 2(\frac{7y}{9} + y) = 260 \)
\( \frac{14y}{9} + 2y = 260 \)
To simplify the equation further, we can multiply through by 9 to eliminate the fractions:
\( 14y + 18y = 2340 \)
\( 32y = 2340 \)
Now, solve for 'y':
\( y = \frac{2340}{32} \)
\( y \approx 73.125 \)
Since the length and width cannot be in decimal form, we need to find a whole number ratio by rounding. Let's round 'y' down to 73. Therefore, the width of the second rectangle is 73 ft.
Now, substitute the value of 'y' into the equation for 'x' to find the length of the second rectangle:
\( x = \frac{7(73)}{9} \)
\( x \approx 63.778 \)
Similarly, we round 'x' down to 63. Therefore, the length of the second rectangle is 63 ft.
Hence, the dimensions of the second rectangle are approximately 63 ft by 73 ft.