Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

In the diagram,
P is the point of intersection of the curves
y=e^(2x) and
y=e^(2-x).
(i) Find the
x-coordinate of
P.
Hence evaluate, correct to two decimal places,
(ii) the area of the shaded region,
(iii) the area of the region bounded by the two curves and the
y-axis.

$13$ . $\newline$ In the diagram, $P$ is the point of intersection of the curves $y=\mathrm{e}^{2 x}$ and $y=\mathrm{e}^{2-x}$ . $\newline$ (i) Find the $x$ -coordinate of $P$ . $\newline$ Hence evaluate, correct to two decimal places, $\newline$ (ii) the area of the shaded region, $\newline$ (iii) the area of the region bounded by the two curves and the $y$ -axis.

View step-by-step help

View step-by-step help

Area of quadrilaterals and triangles: word problems

Full solution

Q.

13

.

\newline

In the diagram,

P

is the point of intersection of the curves

y=\mathrm{e}^{2 x}

and

y=\mathrm{e}^{2-x}

.

\newline

(i) Find the

x

-coordinate of

P

.

\newline

Hence evaluate, correct to two decimal places,

\newline

(ii) the area of the shaded region,

\newline

(iii) the area of the region bounded by the two curves and the

y

-axis.

Set Equations Equal: To find the $x$ -coordinate of $P$ , set the two equations equal to each other because at point $P$ , the $y$ -values of both curves are the same. $\newline$ $e^{2x} = e^{2-x}$
Take Natural Logarithm: Solve for $x$ by taking the natural logarithm ( $\ln$ ) of both sides. $\newline$ $\ln(e^{2x}) = \ln(e^{2-x})$
Use Logarithm Property: Use the property of logarithms that $\ln(e^a) = a$ . $2x = 2 - x$
Isolate Terms with x: Add $x$ to both sides to isolate terms with $x$ on one side. $\newline$ $2x + x = 2$
Combine Like Terms: Combine like terms. $3x = 2$
Divide by $3$ : Divide both sides by $3$ to solve for $x$ . $x = \frac{2}{3}$
Find Area of Shaded Region: Now, to find the area of the shaded region, we need to integrate the difference between the two functions from the $x$ -coordinate of $P$ to the point where the curves meet the $y$ -axis. $\newline$ However, we have not yet found the $x$ -coordinate where the curves meet the $y$ -axis. This is a mistake because we need these $x$ -coordinates to set the limits of integration.

More problems from Area of quadrilaterals and triangles: word problems

Question

Skyler climbs a watchtower to guard against forest fires.

\newline

R(h)=\sqrt{13h}

gives the distance, in kilometers, that Skyler can see when their eyes are

h

meters above the ground.

\newline

A(d)=\pi d^{2}

gives the area, in square kilometers, that Skyler can guard when they can see a distance of

d

\newline

Which expression models the area Skyler can guard when their eyes are

h

meters above the ground?

\newline

1

\newline

13\pi h

\newline

13\pi h^{2}

\newline

\sqrt{13\pi h^{2}}

\newline

169\pi h^{2}

Posted 2 months ago

Question

After the outbreak of a mysterious disease, the average wait time at a hospital increased by

70\%

. Before the outbreak, the average wait time at the hospital was

m

\newline

Which of the following expressions could represent the average wait time at the hospital after the outbreak?

\newline

2

\newline

0.7m

\newline

\frac{170}{100}m

\newline

m+70

\newline

m+0.7

\newline

1.7m

Posted 2 months ago

Question

The radius of a semicircle is

3

yards. What is the semicircle's area?

\newline

\pi \approx 3.14

and round your answer to the nearest hundredth.

\newline

Posted 3 months ago

Question

\newline

A man has bent a circular wire of radius

49\text{cm}

to form a square. Find the sides of a square.

\newline

A)100\text{cm}

\newline

B)50\text{cm}

\newline

C)98\text{cm}

Posted 3 months ago

Question

A quarter is worth

\$0.25

and a half dollar is worth

\$0.50

\newline

a. A quarter has a diameter of

\frac{15}{16}

inch and a height of

\frac{1}{16}

inch. Find the surface area of a quarter. Round your answer to the nearest hundredth.

\newline

b. A half dollar has a diameter of

\frac{9}{8}

inches and a height of

\frac{3}{32}

inch. Find the surface area of a half dollar. Round your answer to the nearest hundredth.

Posted 3 months ago

Question

The hypotenuse of a right triangle measures

17 \mathrm{~cm}

and one of its legs measures

8 \mathrm{~cm}

. Find the measure of the other leg. If necessary, round to the nearest tenth.

\newline

\square

\mathrm{cm}

Posted 3 months ago

Question

One of the legs of a right triangle measures

15 \mathrm{~cm}

and its hypotenuse measures

17 \mathrm{~cm}

. Find the measure of the other leg. If necessary, round to the nearest tenth.

\newline

\square

\mathrm{cm}

Posted 3 months ago

Question

One of the legs of a right triangle measures

8 \mathrm{~cm}

and the other leg measures

15 \mathrm{~cm}

. Find the measure of the hypotenuse. If necessary, round to the nearest tenth.

\newline

\square

\mathrm{cm}

Posted 3 months ago

Question

Find the volume of a right circular cone that has a height of

7.4 \mathrm{~cm}

and a base with a radius of

8.1 \mathrm{~cm}

. Round your answer to the nearest tenth of a cubic centimeter.

\newline

\mathrm{cm}^{3}

Posted 3 months ago

Question

Find the volume of a right circular cone that has a height of

13.3 \mathrm{ft}

and a base with a diameter of

9.4 \mathrm{ft}

. Round your answer to the nearest tenth of a cubic foot.

\newline

\mathrm{ft}^{3}

Posted 3 months ago