So, I have a bunch of math questions and if someone can answer some or all of them that would be great.Question 1:A sandwich box is shown with a right triangle side. The right triangle has hypotenuse 12 cm and base 5 cm.What is the approximate minimum height of a shelf in which this box can fit in the position shown in the picture?A) 4.1 centimetersB) 7.0 centimetersC) 8.5 centimetersD) 10.9 centimetersQuestion 2: Which of the following shows the length of the third side, in inches, of the triangle?One side of the triangle is labeled as 74 inches. The height of the triangle is labeled as 24 inches.A) √6052 inchesB) 50 inchesC) 70 inchesD) √98 inchesQuestion 3:A right triangle PQR is shown with the right angle at Q. The length of side PQ is labeled as 36 cm, and the length of side QR is labeled as 48 cm.What is the length (in centimeters) of side PR of the triangle?A) 36 centimetersB) 42 centimetersC) 60 centimetersD) 84 centimetersQuestion 4:A right triangle with hypotenuse x and legs 7 and 24. What is the length of the unknown side?A) 7B) 24C) 25D) 31Question 5:Three squares having side lengths 6 cm, x cm, and 8 cm joining to form a right triangle at the center. What is the value of x?A) 2 centimetersB) 6 centimetersC) 8 centimetersD) 10 centimetersQuestion 6:The length of the hypotenuse of a right triangle is 145 units. The length of one leg of the triangle is 144. Mike wrote the following step to find the length of the unknown leg:Length of the unknown leg = 145² − 144² = 21,025 − 20,736 = 289 unitsWhich statement best explains whether Mike's step is correct or incorrect?A) It is incorrect because the length of the unknown side is the square root of 289.B) It is incorrect because the length of the unknown side is the square root of 41,761.C) It is correct because the length of the unknown side is the difference of the lengths of the sides.D) It is correct because the length of the unknown side is the difference of the squares of the sides.Question 7:A rectangle has a length of 63 m, a width of x m, and a diagonal of 65 m. What is the value of x?A) 2 metersB) 16 metersC) 63 metersD) 65 metersQuestion 8:A cone is shown with width 2 inches and height 3 inches, and a cylinder is shown with width 2 inches and height 7 inches.How many more cubic inches of juice will cup B hold than cup A when both are completely full? Round your answer to the nearest tenth.A) 18.8 cubic inchesB) 21.9 cubic inchesC) 25.1 cubic inchesD) 32.6 cubic inches

ANSWER TO QUESTION 1.

We use the Pythagoras Theorem to determine  the height of the shelf.

Let [tex]h[/tex] be the height of the triangle,[tex]b[/tex] the base and [tex]c[/tex] the hypotenuse.

Then by the Pythagoras Theorem,

[tex]h^2+b^2=c^2[/tex]

We substitute the base, [tex]b=5[/tex] and the hypotenuse [tex]c=12[/tex]

[tex]h^2+5^2=12^2[/tex]

[tex]h^2+25=144[/tex]

[tex]h^2=144-25[/tex]

[tex]h^2=119[/tex]

[tex]h=\sqrt{119}[/tex]

[tex]h=10.90cm[/tex].

Therefore the approximate minimum height of the shelf should be [tex]h=10.90cm[/tex].

the correct answer is A

ANSWER TO QUESTION 2

We apply the Pythagoras Theorem to find the length of the third side.

See diagram

Let the length of the third side be [tex]y[/tex].

Then

[tex]y^2+24^2=74^2[/tex]

We can now solve for y.

[tex]y^2+576=5476[/tex]

[tex]y^2=5476-576[/tex]

[tex]y^2=4900[/tex]

[tex]y=\sqrt{4900}[/tex]

[tex]y=70[/tex]

The correct answer is C

ANSWER TO QUESTION 3

We use the Pythagoras Theorem to find the length of PR.

Since PR is the hypotenuse .

[tex]|PR|^2=|PQ|^2+|RQ|^2[/tex]

[tex]|PR|^2=36^2+48^2[/tex]

[tex]|PR|^2=1296+2304[/tex]

[tex]|PR|^2=3600[/tex]

[tex]|PR|=\sqrt{3600}[/tex]

[tex]|PR|=60cm[/tex]

The correct answer is C

See diagram in attachment.

ANSWER TO QUESTION 4

The unknown length is the variable [tex]x[/tex], which is the hypotenuse of the right angle triangle.

So we use the Pythagoras theorem to find the unknown length.

[tex]x^2=24^2+7^2[/tex]

[tex]\Rightarrow x^2=576+49[/tex]

[tex]\Rightarrow x^2=625[/tex]

[tex]\Rightarrow x=\sqrt{625}[/tex]

[tex]\Rightarrow x=25[/tex]

The correct answer is C

ANSWER TO QUESTION 5.

From Pythagoras Theorem, the area of the bigger square is equal to the area of the two smaller squares added together.

See diagram in attachment.

That is [tex]x^2=6^2+8^2[/tex].

This implies that,

[tex]x^2=36+64[/tex]

[tex]x^2=100[/tex]

[tex]x=\sqrt{100}[/tex]

[tex]x=10cm[/tex]

The correct answer is D

ANSWER TO QUESTION 6

Let [tex]a[/tex] be the length of the unknown leg.

Then from the Pythagoras Theorem,

[tex]a^2+144^2=145^2[/tex]

This implies that;

[tex]a^2=145^2-144^2[/tex]

[tex]a^2=21,025-20,736[/tex]

[tex]a^2=289[/tex]

[tex]a=\sqrt{289}[/tex]

[tex]a=17 units[/tex]

The correct answer is option A.

It is incorrect because the length of the unknown side is [tex]\sqrt{289}[/tex] and not [tex]289[/tex].

ANSWER TO QUESTION 7

The diagonal is the hypotenuse of the right angle triangle created by the diagonal, the width and the length of the rectangle.

Since the diagonal is the hypotenuse and the two shorter sides are the width and the length of the rectangle, we can apply the Pythagoras Theorem to find the value of [tex]x[/tex].

[tex]x^2+63^2=65^2[/tex]

[tex]x^2+3969=4225[/tex]

[tex]x^2=4225-3969[/tex]

[tex]x^2=256[/tex]

[tex]x=\sqrt{256}[/tex]

[tex]x=16[/tex]

The correct answer is B.

ANSWER TO QUESTION 8

Since the width of the cups is 2 inches, it means the radius is half the width.

That is [tex]r=1[/tex] inch

The volume of a cylinder is given by;

[tex]V=\pi r^2 h[/tex]

The cup with the cylindrical shape (B) will hold

[tex]=1^2\times 7 \pi[/tex]

[tex]=7 \pi[/tex] cubic inches of juice

The volume of a cone is:

[tex]V=\frac{1}{3} \pi r^2 h[/tex]

The cup with the conical shape cup(A), will hold

[tex]V=\frac{1}{3}\times 1^2 \times 3 \pi[/tex]

[tex]V=\pi[/tex]cubic inches of juice

Hence cup B will hold [tex]7\pi -\pi=6\pi=18.8[/tex] cubic inches than cup A.

The correct answer is A