510.
Solve the system of equations: {17
�
−14
�
=5115
�
+16
�
=192
{
17
x
−14
y
=5115
x
+16
y
=192
. - Answer
: �
=9,
�
=6
x
=9,
y
=6
.
511.
Calculate the value of �
x
in the equation log2(
�
+90)−log2(
�
−85)=14log
2
(
x
+90)−log
2
(
x
−85)=14
. - Answer
: �
=86
x
=86
.
512.
Find the area of a regular 680-gon inscribed in a circle with a radius of 3838
units. - Answer
: Area = 680×12×38×38×sin(360
∘
680)680×
21
×38×38×sin
(
680360
∘
)
.
513.
Determine the value of �
x
in the equation 50
�
−2
�
=112589990684262350
x
−2
x
=1125899906842623
. - Answer
: �
=21
x
=21
.
514.
Find the sum of the first 570570
terms of an arithmetic sequence
where the first term is 500500
and the common difference is 380380
. - Answer
: Sum = �
2(2
�
+(
�
−1)
�
)
2
n
(2
a
+(
n
−1)
d
)
with �
=570
n
=570
,
�
=500
a
=500
, �
=380
d
=380
.
515.
Solve the equation 100
�
−2+103
�
+3=106
�
2−
�
−6
x
−2100
+
x
+3103
=
x
2
−
x
−6106
. - Answer
: �
=106
x
=106
.
516.
Calculate the area enclosed by the curve �
=sin(64
�
)
y
=sin(64
x
)
, the x-axis, and the lines �
=0
x
=0
and �
=
�
128
x
=
128
π
. - Answer
: Area = ∫0
�
128sin(64
�
)
��∫
0
128
π
sin(64
x
)
dx
.
517.
Solve the inequality �
2−130
�
+4225<0
x
2
−130
x
+4225<0
. - Answer
: �
=65
x
=65
.
518.
Find the derivative of �
(
�
)=102
�
2+88
�
+102
f
(
x
)=
x
2
+88
x
+102102
.
- Answer
: �
′(
�
)
f
′
(
x
)
.
519.
Determine the value of �
x
if log360(
�
−78)+log360(
�
)=2log
360
(
x
−78)+log
360
(
x
)=2
. - Answer
: �
=720
x
=720
.
