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otez555 [7]
3 years ago
10

To create a giant gemstone, sara first made two identical square pyramids that each had a base area of 100 square inches. Then s

he glued the pyramids' bases together to form the gemstone. The surface area of the gemstone is 520 square inches. What is the value of x? Explain.
Mathematics
1 answer:
inysia [295]3 years ago
5 0

Answer:

8 inches.

Step-by-step explanation:

From the statement we have that they first made two identical square pyramids, each with a base area of 100 square inches.

Ab = s ^ 2 = 100

Therefore each side would be:

s = (100) ^ (1/2)

s = 10

So, side of the square base = 10 inches

Then they tell us that they glued the bases of the pyramids together to form the precious stone. The surface area of the gemstone is 520 square inches, so for a single pyramid it would be:

Ap = 520/2 = 260

For an area of the square pyramid we have the following equation:

Ap = 2 * x * s + s ^ 2

Where x is the height of each triangular surface and s is the side of the square base

Replacing we have:

260 = 2 * x * 10 + 10 ^ 2

20 * x + 100 = 260

20 * x = 160

x = 160/20

x = 8

Therefore, the value of x is 8 inches.

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omeli [17]

Problem 18

Answers:

~P -> Q: "if it is not fall, then the leaves change colors"

~P -> ~Q: "if it is not fall, then the leaves don't change colors"

P -> ~Q: "If it is fall, then the leaves don't change colors"

Q -> ~P: "If the leaves change colors, then it is not fall"

----------

Explanation:

Simply replace each letter with the corresponding definition. We replace each P with "it is fall" and each ~P with "it is not fall". The squiggly line means "not". The same idea applies with Q and ~Q.

==========================================

Problem 19

Answer: Choice A) 134

----------

Explanation:

AB = CD

8x = 4x+24

8x-4x = 24

4x = 24

x = 6

BC = 18x+26

BC = 18*6+26

BC = 134

==========================================

Problem 20

Answer: Choice C) x = 5 and y = 11

----------

Explanation:

(anglePQR)+(angleRQS) = 90

(12x-2)+(6x+2) = 90

18x = 90

x = 5

angle TQU = 90

10y-20 = 90

10y = 110

y = 11

==========================================

Problem 21

Answer:  99

----------

Explanation:

The remote interior angles add to the exterior angle (remote interior angle theorem)

(angle RPQ) + (angle PQR) = angle PRS

(3x-29) + (7x-42) = 6x-3

10x-71 = 6x-3

10x-6x = -3+71

4x = 68

x = 68/4

x = 17

angle PRS = 6x-3

angle PRS = 6*17-3

angle PRS = 99

==========================================

Problem 22

Answer: 116

----------

Explanation:

Refer to the diagram below.

Solve 105+x = 180 to get x = 75

Solve 135+y = 180 to get y = 45

Solve x+y+z = 180 to get z = 60

Solve z+82+p = 180 to get p = 38

Solve p+78+q = 180 to get q = 64

Solve q+r = 180 to get r = 116

==========================================

Problem 23

Answer: B) between 4 and 44 inches

----------

Explanation:

We use a variation of the triangle inequality theorem here.

If p is the missing side length, then

24-20 < p < 24+20

4 < p < 44

So,

4 < XY < 44

The side XY cannot be 4 inches and it cannot be 44 inches either.

==========================================

Problem 24

4 Answers: A, B, C, D

----------

Explanation:

Same idea as problem 23

24-10 < x < 24+10

14 < x < 34

So we'll select values between 14 and 34, excluding each endpoint. Those values are x = 16, 18, 24, 26

In other words, only 12 and 34 are non-answers.

==========================================

Problem 25

Answer: C) 84 degrees

----------

Explanation:

The given exterior angle is 152 degrees

Subtract that from 180 to get the adjacent interior angle 180-152 = 28

Now let's find the missing angle of triangle DEB

D+E+B = 180

28+68+B = 180

96+B = 180

B = 180-96

B = 84

This is the measure of angle DBE

This is also the measure of angle ABC because the two angles are vertical angles.

Vertical angles are always congruent.

4 0
3 years ago
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