### Still have math questions?

Ask our expert tutors
Q:

For the same 18-foot ladder, how far up on the house will the ladder reach?

Q:

The measure of one angle of a right triangle is $$24 ^ { \circ }$$ more than the measure of the smallest angle. Find the measures of all three angles, and separate your answers with commas.

Q:

The measure of one angle of a right triangle is $$24 ^ { \circ }$$ more than the measure of the smallest angle. Find the measures of all three angles, and separate your answers with commas.

Q:

Points $$A$$ and $$B$$ are on opposite sides of a lake. A point $$C$$ is $$81.3$$ meters from $$A$$ . The measure of angle BAC is $$78.33 ^ { \circ } ,$$ and the measure of angle $$A C B$$ is determined to be $$34.167 ^ { \circ }$$ . Find the distance between points $$A$$ and $$B$$ (to the nearest meter).

A. $$35 m$$ B. $$25 m$$ C. $$54 m$$ D. $$49 m$$

Q:

Your uncle Mike's boat can travel $$4.0 m / s$$ in still water. One sunny afternoon, you and Unk Mike decide to go fishing. While waiting for a bite, you begin thinking, "If this river is flowing at $$5.5 m / s$$ southward, and we are heading eastward, directly across the river, what are the direction and magnitude of our total velocity?" Answer your own question.

Q:

You are hired for a summer job painting the exterior of houses for the elderly in your community. You have completed your on-the-job training and are set to begin your first painting job today. Before you leave, your manager reminds you of the most important rule: Ladders must make an angle of between $$40$$ and $$70$$ degrees with the ground to ens proper safety.

Upon arriving at the home, you unload your $$12$$ - and $$18$$ -foot ladders and begin the preparatlions for painting. Use your information on $$45 ^ { \circ } - 45 ^ { \circ } - 90 ^ { \circ }$$ and $$30 ^ { \circ } - 60 ^ { \circ } - 90 ^ { \circ }$$ right triangles to answer the following questions. Round your answers to the nearest tenth an please show all your work.

1. You wish to use your $$12$$ -foot ladder to reach the lower level of the home. How far away from the home must the ladder be placed to form a $$45 ^ { \circ } - 45 ^ { \circ } - 90 ^ { \circ }$$ triangle?

Q:

The area. of a triangle is $$1374.74$$ square meters. We know two of its sides are $$75$$ meters and $$133$$ meters. Use the area formula for triangles shown below to determine the measure of the included angle between these two sides.

$$A = \frac { 1 } { 2 } a \cdot b \circ \sin C$$

Q:

Describe the relationship between the ordered pairs for each point on the unit circle arid the primary trigonometric ratios.

In which quadrants is the sine ratio positive? Cosine? Tangent? Why do you suppose that is?

For any chosen point on the unit circle, how many other points will have the same y-coordinate? Explain why.

In the first quadrant, $$45 ^ { \circ }$$ has a sine of $$\frac { 1 } { \sqrt { 2 } }$$ . Where else is this true? Why do you suppose that is?

In the first quadrant, $$45 ^ { \circ }$$ has a cosine of $$\frac { 1 } { \sqrt { 2 } }$$ . Where else is this true? Why do you suppose that is? In the first quadrant, $$45 ^ { \circ }$$ has a tangent of $$\frac { 1 } { 1 }$$ , or $$1$$ . Where else is this true? Why do you suppose that is?

Does that same relationship exist for other angles? Why do you suppose that is?

Q:

In which quadrants is the sine ratio positive? Cosine? Tangent? Why do you suppose that is?

Q:

If $$\sin \frac { \pi } { 12 } = \frac { 1 } { 2 } \sqrt { A - \sqrt { B } } ,$$ then, by using a half-angle formula, find

$$A =$$

$$B =$$

Q:

1. When flying in a plane over Prince Edward Island it is possible to see all the way across the island from one side to the other. The pilot of a small plane spots the western edge of the island at an angle of depression of $$61 ^ { \circ }$$ to and the eastern edge of the island with an angle of depression of $$3 ^ { \circ }$$ . If the plane is flying at an altitude of $$10.5 km$$ , how long is Prince Edward Island?

Q:

A rabbit is hopping along a garden path at the same time that a bird takes off from the ground and travels $$425 m$$ in a northeasterly direction. What is the angle of elevation between the bird and the rabbit after the rabbit has hopped $$320 m$$ ?

____

Q:

When the sun's angle of elevation is $$38$$ degrees, a building casts a shadow of $$45 m$$ . How high is the building?

____meters

Q:

An advertising blimp hovers over the football stadium in Detroit, where a very lovely geometry teacher is watching the Green Bay Packers play the Detroit Lions. The pilot of the blimp sights the very lovely geometry teacher at an angle of depression of $$8$$ degrees. Find the distance between the pilot and the very lovely geometry teacher, if the pilot is waving from an altitude of $$125 m$$ .

____meters

Q:

A person who is $$1.5 m$$ tall is standing $$20 m$$ from the base of a building. The person sights the top of the building with an angle of elevation of $$58$$ degrees. Find the height of the building to the nearest meter. (Hint: think about what you need to do with the person's height.)

____meters

Q:

What is the distance from Kareem to the house?

Kareem and Amy are standing on a riverbank, $$150$$ meters apart, at points $$A$$ and $$B$$ respectively. (See the figure below.) They want to know the distance from Kareem to a house located across the river at point $$C$$ . Kareem measures angle $$A$$ (angle $$B A C$$ ) to be $$55 ^ { \circ }$$ , and Amy measures angle $$B$$ (angle $$A B C$$ ) to be $$74 ^ { \circ }$$ . What is the distance from Kareem to the house? Round your answer to the nearest tenth of a meter.

Q:

Suppose a projectile is fired from a cannon with velocity $$v _ { 0 }$$ and angle of elevation $$\theta$$ . The horizontal distance $$R ( \theta )$$ it travels (in feet) is given by the following.

$$R ( \theta ) = \frac { ( v _ { 0 } ) ^ { 2 } \sin 2 \theta } { 32 }$$

If $$v _ { 0 } = 80 ft$$ /s, what angle $$0$$ (in radians) should be used to hit a target on the ground $$118$$ feet in front of the cannon?

Q:

Consider the derivation of an alternate form of the cosine double angle identity.

1. $$\cos ( 2 x ) = \cos ^ { 2 } ( x ) - \sin ^ { 2 } ( x )$$

2. $$= \cos ^ { 2 } ( x ) - ( 1 - \cos ^ { 2 } ( x ) )$$

3. $$= \cos ^ { 2 } ( x ) - 1 - \cos ^ { 2 } ( x )$$

4. $$= 2 \cos ^ { 2 } ( x ) - 1$$

What is the error in this derivation?

In step $$1 , \cos ( 2 x )$$ is equal to $$\cos ^ { 2 } ( x ) + \sin ^ { 2 } ( x )$$ .

In step $$2 , \sin ^ { 2 } ( x )$$ should have been replaced with $$1 +$$ $$\cos ^ { 2 } ( x )$$ .

In step $$3 , \cos ^ { 2 } ( x ) - 1 - \cos ^ { 2 } ( x )$$ should be $$\cos ^ { 2 } ( x ) - 1$$ $$+ \cos ^ { 2 } ( x )$$

In $$\operatorname { step } 4 , 2 \cos ^ { 2 } ( x ) - 1$$ should be $$1 - 2 \cos ^ { 2 } ( x )$$

Q:

Suppose $$A B C$$ is a right triangle with sides a, b, and c and right angle at C. Use the Pythagorean theorem to find the unknown s the denominators when apolicable. nide length. Then find the values of the six trigonometric functions for angle B. Rationalize the denominators when applicable. $$a = 3 , b = 4$$

What is the length of side c? (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

$$\sin B = ?$$

Q:

Write the ratios for $$\sin M , \cos M ,$$ and $$\tan M$$ . Give the exact value and a four-decimal approximation.

$$\sin M = \square$$

(Type an exact answer in simplified form. Type an integer or a fraction.) Type the decimal approximation of the answer rounded to four decimal places.

$$\sin M = \square$$

(Round the final answer to four decimal places as needed.)

Q:

You need to find the distance across a river, so you make a triangle. BC is $$943$$ feet, m $$\angle B = 102.9 ^ { \circ }$$ and $$m \angle C = 18.6 ^ { \circ }$$ . Find AB.

A. $$353 ft$$

B. $$299 ft$$

C. $$356 ft$$

D. $$188 ft$$

Q:

Triangle $$A B C$$ has side lengths $$9,40 ,$$ and $$41 .$$ Do the side lengths form a Pythagorean triple? Explain.

a Yes, they form a Pythagorean triple; $$9 ^ { 2 } + 40 ^ { 2 } = 41 ^ { 2 }$$ and $$9,40$$ , and $$41$$ are all nonzero whole numbers.

b No, they do not form a Pythagorean triple; $$9 ^ { 2 } + 40 ^ { 2 } \neq 41 ^ { 2 }$$ .

c No, they do not form a Pythagorean triple; although $$9 ^ { 2 } + 40 ^ { 2 } = 41 ^ { 2 }$$ , the side lengths do not meet the other requirements of a Pythagorean triple.

d.Yes; they can form a right triangle, so they form a Pythagorean triple.

Q:

A triangle has side lengths of $$14 cm , 48 cm$$ , and $$50 cm$$ . Classify it as acute, obtuse, or right.

a. obtuse

b. acute

c. right

Q:

A radio tower is $$100$$ feet high. A support cable $$140$$ feet long is fastened to the top of the tower and is anchored in the ground. How far from the base of the tower will the cable be anchored? Give your answer to the nearest foot.

$$40$$ feet

$$98$$ feet

$$120$$ feet

$$172$$ feet

Q:

A kite is on a string as shown in the figure below.

The string makes an angle of $$60 ^ { \circ }$$ with the ground. If the length of the string is $$120$$ feet, what is the height of the kite above the ground, in feet?

$$60$$

$$60 \sqrt { 3 }$$

$$120$$

$$120 \sqrt { 3 }$$

Q:

Josh tied a helium-filled balloon to a stake in the ground. The wind blew it so that the $$10$$ -foot string attached to the balloon made an angle of $$45 ^ { \circ }$$ with the ground.

What is the approximate height of the balloon from the ground?

Q:

Brad stands $$30$$ feet from a tree. He estimates the angle of elevation from a point on the ground $$30$$ feet from the tree to the top of the tree to be $$60 ^ { \circ }$$ as shown below.

Which of the following is closest to the height of the tree?

Q:

10 -foot ladder is placed against a building so that the ladder makes a $$78 ^ { \circ }$$ angle with the ground. To the nearest tenth of a foot, at what height does the ladder touch the building?

7.8 feet

9.8 feet

10.2 feet

16.2 feet

Q:

Find the measures of the angles of $$\triangle G H J$$ , where  $$\overline { G H }$$ is a diameter of $$O 0$$ , and chord $$\overline { G J }$$ intercepts an arc with a measure of $$70$$ .

After you enter your answer press GO.

$$m \angle J =$$

$$m \angle H =$$

$$m \angle G =$$

Q:

Garrett is painting a prop for his school play. The prop is a rectangle that measures $$5.75$$ feet by $$7.25$$ feet. About how much area will he need to paint?

$$13 ft ^ { 2 }$$

$$42 ft ^ { 2 }$$

C  $$11 ft ^ { 2 }$$

$$21 ft ^ { 2 }$$

Q:

Because the values of circular functions repeat every $$2 \pi$$ , they are used to describe things that repeat periodically. For example, the maximum afternoon temperature in a given city might be modeled by the formula below. In the formula, t represents the maximum afternoon temperature in month $$x$$ , with $$x = 0$$ representing January, $$x = 1$$ representing February, and so on.

$$t = 15 - 10 \cos \frac { x \pi } { 6 }$$

The maximum afternoon temperature in January is ___. (Round to the nearest integer as needed.)

Q:

Because the values of circular functions repeat every $$2 \pi$$ , they are used to describe things that repeat periodically. For example, the maximum afternoon temperature in a given city might be modeled by the formula below. In the formula, t represents the maximum afternoon temperature in month $$x$$ , with $$x = 0$$ representing January, $$x = 1$$ representing February, and so on.

$$t = 15 - 10 \cos \frac { x \pi } { 6 }$$

The maximum afternoon temperature in January is (Round to the nearest integer as needed.)

In May, the maximum afternoon temperature is (Round to the nearest integer as needed.)

Q:

The base of a triangle exceeds the height by $$7$$ centimeters. If the area is $$114$$ square centimeters, find the length of the base and the height of the triangle.

Q:

Use similar triangles to solve. A person who is $$6$$ feet tall is standing $$132$$ feet from the base of a tree, and the tree casts a $$143$$ foot shadow. The person's shadow is

$$11$$ feet in length. What is the height of the tree?

Q:

Because the values of circular functions repeat every $$2 \pi$$ , they are used to describe things that repeat periodically. For example, the maximum afternoon temperature in a given city might be modeled by the formula below. In the formula, t represents the maximum afternoon temperature in month $$x$$ , with $$x = 0$$ representing January, $$x = 1$$ representing February, and so on.

$$t = 15 - 10 \cos \frac { x \pi } { 6 }$$

The maximum afternoon temperature in January is ___(Round to the nearest integer as needed.)

In May, the maximum afternoon temperature is___ . (Round to the nearest integer as needed.)

The maximum afternoon temperature in September is ___ (Round to the nearest integer as needed.)

Q:

Because the values of circular functions repeat every $$2 \pi$$ , they are used to describe things that repeat periodically. For example, the maximum afternoon temperature in a given city might be modeled by the formula below. In the formula, t represents the maximum afternoon temperature in month $$x = 0$$ representing January, $$x = 1$$ representing February, and so on.

$$t = 15 - 10 \cos \frac { x } { 6 }$$

The maximum afternoon temperature in January is $$5 ^ { \circ } .$$

(Round to the nearest integer as needed.)

In May, the maximum afternoon temperature is $$20 ^ { \circ } .$$

(Round to the nearest integer as needed.)

The maximum afternoon temperature in September is $$\square ^ { 0 } .$$

(Round to the nearest integer as needed.)

Q:

Right $$\triangle J K L$$ has a hypotenuse of length $$34$$ feet and one leg of length $$16$$ feet. What is the area of $$\triangle J K L$$ ?

Q:

If the vertex angle of an isosceles triangle is

$$60 ^ { \circ }$$ and the length of each leg is $$26$$ , what is the length of the base?

Q:

In $$\triangle C D E , m \angle C$$ is twice $$m \angle D$$ and $$m \angle E$$ is $$3$$ times $$m \angle D$$ . If $$C D = 16 ,$$ what is $$D E ?$$

Q:

What is the length of the arc that subtends a central angle of $$82 ^ { \circ }$$ for a circle with a radius of $$9$$ centimeters? Use $$3.14$$ for

$$\pi$$ when necessary.

A. $$12.87 cm$$

B. $$6.56 cm$$

C. $$14.60 cm$$

D. $$12.60 cm$$

Q:

What is the area of the sector having a radius of $$8$$ and a central angle of $$\frac { 5 \pi } { 3 }$$ radians?

A. $$\frac { 320 \pi } { 3 }$$ units $$^ { 2 }$$

B. $$\frac { 140 \pi } { 3 }$$ units $$^ { 2 }$$

C. $$50 \pi$$ units $$^ { 2 }$$

D. $$\frac { 160 \pi } { 3 }$$ units $$^ { 2 }$$

Q:

Write the following function in terms of its cofunction. Assume that all angles in which an unknown appears are acute angles.

$$\cot ( \theta - 15 ^ { \circ } )$$

A. $$\tan ( \theta - 105 ^ { \circ } )$$

B. $$\tan ( 15 ^ { \circ } - \theta )$$

C. $$\tan ( 105 ^ { \circ } - \theta )$$

D. $$\cot ( 105 ^ { \circ } - \theta )$$

Q:

Solve the right triangle if angle $$B = 50.1 ^ { \circ }$$ and side $$a = 121$$ inches. Assume that $$C = 90 ^ { \circ }$$ A. $$A = 50.1 ^ { \circ } , b = 101 , c = 111$$

B. $$A = 39.9 ^ { \circ } , b = 145 , c = 111$$

C. $$A = 39.9 ^ { \circ } , b = 145 , c = 189$$

D. $$A = 39.9 ^ { \circ } , b = 101 , c = 189$$

Q:

Evaluate the following trigonometric function at the quadrantal angle, or state that the expression is undefined.

$$\sin \frac { 3\pi } { 2 }$$

Q:

Find the exact value of each of the remaining trigonometric functions of $$\theta .$$

$$\cos \theta = - \frac { 5 } { 13 } , \theta$$ in Quadrant III

Q:

Let $$\theta$$ be an angle in standard position. Name the quadrant in which $$\theta$$ lies.

$$\tan \theta < 0 , \csc \theta > 0$$

Q:

In a unit circle, the radian measure of the central angle is equal to the length of the ___

Choose the correct answer below.

A. radius of the circle

B. intercepted arc

C. diameter of the circle

Q:

A plane rises from take-off and flies at an angle of $$5 ^ { \circ }$$ with the horizontal runway. When it has gained $$500$$ feet, find the distance, to the nearest foot, the plane has flown

The.plane has flown approximately $$\square$$ feet. (Round to the nearest foot.)

Q:

Use the $$( x , y )$$ coordinates in the figure to find the value of tan $$2 \pi$$ or state that the expression is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. $$18$$ of $$20,20$$ point(s) possible

A. tan $$2 \pi =$$

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the possible expression. Rationalize the denominator. )

B. The expression is undefined.

Q:

Use the $$( x , y )$$ coordinates in the figure to find the value of the trigonometric function at the indicated real number, $$t$$ , or state that the expression is undefined.

Q:

Use the triangle shown to the right to evaluate the following expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

$$\sin 45 ^ { \circ }$$

Q:

Use the given triangles to evaluate the following expression. If necessary, express the value without a square root in the denominator by rationalizing the denominator.

$$\sec ( \frac { \pi } { 4 } )$$

Q:

Find the exact value of each of the remaining trigonometric functions of $$\theta$$ .

$$\cos \theta = - \frac { 5 } { 13 } , \theta$$ in Quadrant III

$$\sin \theta = \square$$

$$\tan \theta = \square$$

$$\cot \theta = \square$$

$$\sec \theta = \square$$

$$\csc \theta = \square$$

Q:

Use identities to find values of the sine and cosine functions of the function for the angle measure.

$$2 x$$ , given $$\tan x = - 4$$ and $$\cos x > 0$$

$$\cos 2 x = \frac { - 15} { 17}$$

(Simplify your answer, including any radicalsf Use integers or fractions for any numbers in the expression.)

$$\sin 2 x = \square$$

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Q:

A plane at a height of $$5.5 km$$ , approaches the airport (C) at an angle of depression of $$30 ^ { \circ } .$$ How far is thas plane from the airport along its descending flight path, $$x$$ , to the nearest tenth of a kilometre? *

Q:

The triangles DHF and DGE shown below are similar triangles. DE is $$12.5 m , DF$$ is

$$22.5 m$$ , and EG is $$20.0 m$$ . What is the length of $$FH ? ^ { * }$$

Q:

Determine the quadrant where the angle is:

(a) $$106 ^ { \circ }$$

(b) $$219 ^ { \circ }$$

Q:

The angle of elevation to the top of a Building in New York is found to be $$3$$ degrees from the ground at a distance of $$2$$ miles from the base of the building. Using this information, find the height of the building.

Q:

Find an angle $$\theta$$ with $$0 ^ { \circ } < \theta < 360 ^ { \circ }$$ that has the same: Sine as $$250 ^ { \circ } : \theta =$$ degrees Cosine as $$250 ^ { \circ } : \theta =$$

Q:

Find polar coordinates for the point with rectangular coordinates $$( - 2,2 \sqrt { 3 } )$$ if $$0 \leq \theta \leq 2 \pi$$ and $$r \geq 0$$ .

$$( 4 , \frac { 2 \pi } { 3 } )$$

B. $$( 2 , \frac { 2 \pi } { 3 } )$$

C. $$( 4 , \frac { \pi } { 3 } )$$

D. $$( 4 , \frac { 5 \pi } { 6 } )$$

Q:

Which is the angle of elevation from the boat to the lighthouse?

Q:

Which is the angle of depression from the top of the waterfall to Kelley?

Q:

Which is the angle of elewation from dim to the top of the waterfall?

Q:

Write the following in terms of $$\sin \theta$$ and $$\cos \theta$$ ; then simplify if possible. (Leave your answer in terms of $$\sin \theta$$ and/or $$\cos \theta$$ .)

$$\csc \theta \cot \theta$$

Q:

Find the measures of two angles, one positive and one negative, that are coterminal with the given angle.

$$\frac { 9 \pi } { 7 } ; - \frac { 9 \pi } { 7 }$$

$$\frac { 16 \pi } { 7 } ; - \frac { 12 \pi } { 7 }$$

$$\frac { 12 \pi } { 7 } ; - \frac { 16 \pi } { 7 }$$

$$\frac { 9 \pi } { 7 } ; - \frac { 12 \pi } { 7 }$$

Q:

Determine the length of the missing side of a right triangle of the length of one leg is 12 ft and the length of the other leg is 15 ft.

Q:

In the Daylily Garden, $$m \angle A \approx 56 ^ { \circ }$$ and $$m \angle C \approx 58 ^ { \circ }$$ . Which is the shortest side of the Daylily Garden? Explain.

Q:

A car is traveling at $$112 km / h$$ due south at a point $$\frac { 4 } { 5 }$$ kilometer north of an intersection. A police car is traveling at $$96 km / h$$ due west at a point $$\frac { 2 } { 5 } kilometer$$ due east of the same intersection. At that instant, the radar in the police car measures the rate at which the distance between the two cars is changing. What does the radar gun register? Round your answer to three decimal places.

Q:

A triangle is composed of sides of $$16$$ feet, $$41$$ feet, and a third side of unknown length. Select ALL degrees below that could be the length of the third side

$$25$$

$$34$$

$$45$$

$$18$$

$$73$$

$$29$$

Q:

Solve the right triangle, $$\triangle ABC$$ , for the missing side and angles to the nearest tenth given sides

$$a = 13.2$$ and $$b = 17.7$$ .

A. $$A = 41.8 , B = 48.2 , c = 11.8$$

$$B . A = 36.7 , B = 53.3 , c = 22.1$$

C. $$A = 36.7 , B = 53.3 , c = 11.8$$

D. $$A = 48.2 , B = 41.8 , c = 22.1$$

Q:

The entrance of the walkway over the river is parallel to the front entrance of the building. What is the value of $$x$$ ?

Q:

In the triangle below, what is the length of the side opposite the $$60 ^ { \circ }$$ angle?

A. $$3 \sqrt { 3 }$$

B. $$6$$

C. $$\sqrt { 3 }$$

D. $$2 \sqrt { 3 }$$

Q:

Consider a triangle $$A B C$$ like the one below. Suppose that $$C = 99 ^ { \circ } , a = 45$$ , and $$b = 16$$ . (The figure is not drawn to scale.) Solve the triangle. Carry your intermediate computations to at least four decimal places, and round your answers to the nearest tenth.

If there is more than one solution, use the button labeled "or".

Q:

Angle $$\alpha$$ is in quadrant III and angle $$\beta$$ is in quadrant IV. If $$\sin \alpha = - \frac { 5 } { 6 }$$ and $$\cos \beta = \frac { 1 } { 2 }$$ , find $$\cos ( \alpha + \beta )$$ .

$$\frac { \sqrt { 11 } + 5 \sqrt { 3 } } { 6 }$$

$$\frac { \sqrt { 11 } 5 \sqrt { 3 } } { 6 }$$

$$\frac { \sqrt { 11 } 5 \sqrt { 3 } } { 12 }$$

$$\frac { \sqrt { 11 } | 5 \sqrt { 3 } } { 12 }$$

Q:

A triangular field has sides of $$218.5 m$$ and $$213.3 m$$ , and the angle between them measures $$58.96 ^ { \circ }$$ . Find the area of the field.

A. $$14,332 m ^ { 2 }$$

B. $$12,016 m ^ { 2 }$$

C. $$24,032 m ^ { 2 }$$

Q:

$$\triangle A B C$$ is a right triangle with legs measuring $$5$$ centimeters and $$12$$ centimeters. Which of the following is the measure of the triangle's hypotenuse?

A. $$13 cm$$

B. $$10 cm$$

C. $$11 cm$$

D. $$12 cm$$

Q:

Two airplanes leave the airport. Plane $$A$$ departs at a $$44 ^ { \circ }$$ angle from the runway, and plane B departs at a $$40 ^ { \circ }$$ from the runway. Which plane was farther away from the airport when it was $$6$$ miles from the ground? Round the solutions to the nearest hundredth.

Q:

Triangle DEF contains right angle E. If angle D measures $$55 ^ { \circ }$$ and its opposite side measures $$7.6 u$$ what is the measure of side DF? Round your answer to the nearest hundredth.

Q:

Points $$A$$ and $$B$$ are on opposite sides of a lake. A point $$C$$ is $$91.791$$ meters from $$A$$ . The measure of $$\angle B A C$$ is $$76.833 ^ { \circ }$$ , and the measure of $$\angle A C B$$ is determined to be $$31.683 ^ { \circ }$$ . Find the distance between points $$A$$ and $$B$$ .

A. $$71.483 m$$

B. $$50.843 m$$

C. $$165.718 m$$

D. $$108.517 m$$

Q:

Each panel of a window is in the shape of a parallelogram. The measure of one of acute angles is $$50 ^ { \circ }$$ . What is the measure of one of the obtuse angles?

$$140 ^ { \circ }$$

$$130 ^ { \circ }$$

$$110 ^ { \circ }$$

$$120 ^ { \circ }$$

Q:

The altitude of a mountain peak is measured as shown in the figure to the right. At an altitude of $$14,578$$ feet on a different mountain, the straight-line distance to the peak of Mountain $$A$$ is $$27.4823$$ miles and the peak's angle of elevation is $$\theta = 5.6800 ^ { \circ }$$ . Approximate the height (in feet) of Mountain A.

Q:

Because the values of circular functions repeat every $$2 \pi$$ , they are used to describe things that repeat periodically. For example, the maximum afternoon temperature in a given city might be modeled by the formula below. In the formula, trepresents the maximum afternoon temperature in month $$x$$ , with $$x = 0$$ representing January, $$x$$ representing February, and so on.

$$t = 15 - 10 \cos \frac { x \pi } { 6 }$$

The maximum afternoon temperature in January is $$5 ^ { \circ }$$

(Round to the nearest integer as needed.) In May, the maximum afternoon temperature is $$20 ^ { \circ }$$

(Round to the nearest integer as needed.)

The maximum afternoon temperature in September is $$\square ^ { \circ }$$

(Round to the nearest integer as needed.)

Q:

In a right triangle, $$\sin ( \beta ) = 0.292$$ . If the hypotenuse of the triangle has a length of $$17.5 cm$$ ; what is the length of the side adjacent to angle $$\beta$$ . Note: You will also need to use the Pythagorean theorem to answer this question.

$$11.4 cm$$

$$16.7 cm$$

Q:

The solid lies between planes perpendicular to the $$x$$ -axis at $$x = - 1$$ and $$x = 1$$ . The cross-sections perpendicular to the $$x$$ -axis between these planes are squares whose bases run from the semicircle $$y = - \sqrt { 1 - x ^ { 2 } }$$ to the semicircle $$y = \sqrt { 1 - x ^ { 2 } }$$

Q:

Extreme values on a helix Suppose that the partial derivatives of a function $$f ( x , y , z )$$ at points on the helix $$x = \cos t , y = \sin t$$ , $$z = t$$ are $$f _ { x } = \cos t , f _ { y } = \sin t , f _ { z } = t ^ { 2 } + t - 2$$ .

Q:

97. Here is the definition of an infinite right-hand limit. Suppose that an interval $$( c , d )$$ lies in the domain of $$f$$ . We say that $$f ( x )$$ approaches infinity as $$x$$ approaches $$c$$ from the right, and write $$\lim _ { x \rightarrow c ^ { + } } f ( x ) = \infty$$ , $$f ( x ) > B$$ whenever $$c < x < c + \delta$$ if, for every positive real number $$B$$ , there exists a corresponding number $$\delta > 0$$ such that Modify the definition to cover the following cases.

a. $$\lim _ { x \rightarrow c ^ { - } } f ( x ) = \infty$$

b. $$\lim _ { x \rightarrow c ^ { + } } f ( x ) = - \infty$$

c. $$\lim _ { x \rightarrow c ^ { - } } f ( x ) = - \infty$$

Q:

Tilted plate Calculate the fluid force on one side of a $$5 ft$$ by $$5 ft$$ square plate if the plate is at the bottom of a pool filled with water to a depth of $$8 ft$$ and

a. lying flat on its $$5 ft$$ by $$5 ft$$ face.

b. resting vertically on a $$5$$ -ft edge.

c. resting on a $$5$$ -ft edge and tilted at $$45 ^ { \circ }$$ to the bottom of the pool.

Q:

Compute the average value of the temperature function $$f ( x ) = 37 \sin ( \frac { 2 \pi } { 365 } ( x - 101 ) ) + 25$$

Q:

Find an equation for the curve traced by the point $$P ( x , y )$$ if the distance from $$P$$ to the vertex of the parabola $$x ^ { 2 } = 4 y$$ is twice the distance from $$P$$ to the focus. Identify the curve.

Q:

Among all rectangular regions $$0 \leq x \leq a , 0 \leq y \leq b$$ , find the one for which the total outward flux of $$F = ( x ^ { 2 } + 4 x y ) i - 6 y j$$ across the four sides is least. What is the least flux?

Q:

Distance between skew lines Find the distance between the line $$L _ { 1 }$$ through the points $$A ( 1,0 , - 1 )$$ and $$B ( - 1,1,0 )$$ and the line $$L _ { 2 }$$ through the points $$C ( 3,1 , - 1 )$$ and $$D ( 4,5 , - 2 )$$ . The distance is to be measured along the line perpendicular to the two lines. First find a vector $$n$$ perpendicular to both lines. Then project $$\vec { A C }$$ onto $$n$$ .

Q:

Generalizing the Product Rule The Derivative Product Rule gives the formula $$\frac { d } { d x } ( u v ) = u \frac { d v } { d x } + \frac { d u } { d x } v$$ for the derivative of the product $$u v$$ of two differentiable functions of $$x$$ .

a. What is the analogous formula for the derivative of the prod- uct $$u v w$$ of three differentiable functions of $$x$$ ?

b. What is the formula for the derivative of the product $$u _ { 1 } u _ { 2 } u _ { 3 } u _ { 4 }$$ of four differentiable functions of $$x$$ ?

Q:

Lifting a rope A mountain climber is about to haul up a $$50 - m$$ length of hanging rope. How much work will it take if the rope weighs $$0.624 N / m$$ ?

Q:

A formula for the curvature of the graph of a function in the $$x y$$ -plane

a. The graph $$y = f ( x )$$ in the $$x y$$ -plane automatically has the parametrization $$x = x , y = f ( x )$$ , and the vector formula $$r ( x ) = x i + f ( x ) j$$ . Use this formula to show that if $$f$$ is a twice-differentiable function of $$x$$ , then $$\kappa ( x ) = \frac { | f ^ { \prime \prime } ( x ) | } { [ 1 + ( f ^ { \prime } ( x ) ) ^ { 2 } ] ^ { 3 / 2 } }$$ .

Q:

The equation $$x ^ { 2 } = 2 ^ { x }$$ has three solutions: $$x = 2 , x = 4$$ , and one other. Estimate the third solution as accurately as you can by graphing.

Q:

Find the dimensions of a right circular cylinder of maximum vol- ume that can be inscribed in a sphere of radius $$10 cm$$ . What is the maximum volume?

Q:

Sound Location Two microphones, $$2$$ miles apart, record an explosion. Microphone A receives the sound $$6$$ seconds before microphone B. Where did the explosion occur? (Assume sound travels at $$1100$$ feet per second.)

Q:

Finding volume The region in the first quadrant that is enclosed by the $$x$$ -axis and the curve $$y = 3 x \sqrt { 1 - x }$$ is revolved about the $$y$$ -axis to generate a solid. Find the volume of the solid.

Q:

Temperature change It took $$14 sec$$ for a mercury thermometer to rise from $$- 19 ^ { \circ } C$$ to $$100 ^ { \circ } C$$ when it was taken from a freezer and placed in boiling water. Show that somewhere along the way the mercury was rising at the rate of $$8.5 ^ { \circ } C / sec$$ .

Q:

Height of a Ferris Wheel Car A model for the height $$h$$ (in feet) of a Ferris wheel car is $$h = 50 + 50 \sin 8 \pi t$$ where $$t$$ is the time (in minutes). (The Ferris wheel has a radius of $$50$$ feet.) This model yields a height of $$50$$ feet when $$t = 0$$ . Alter the model so that the height of the car is $$1$$ foot when $$t = 0$$ .