in exercises 19 and 20, determine whether ( f(x) = 0 ) has any repeated real solutions. explain your reasoning.
19. graph of a function
20. graph of a function

21. modeling real life archaeologists discovered several huge concrete blocks at the ruins of caesarea. one of the blocks has a volume of 90 cubic meters and the dimensions shown. find the value of ( x ). (see example 3.)
diagram of a rectangular block with dimensions ( (2x - 5) ) m, ( x ) m, ( (2x - 4) ) m

22. modeling real life rivergate tower is the seventh - largest building in tampa. the cylindrical tower has a volume of about ( 56,000pi ) cubic meters and has the radius and height shown (in meters). find the radius and height of the tower.
diagram of a cylinder

23. problem solving you are designing a marble basin that will hold a fountain for a city park. the sides and bottom of the basin should be 1 foot thick. its outer length should be twice its outer width and outer height. what should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?
diagram of a rectangular basin with 1 ft thickness, outer length ( 2x ), outer width ( x ), outer height ( x )

24. how do you see it? the graph of a quartic function ( f ) is shown.
a. what are the real zeros of ( f )?
graph of a quartic function
b. write an equation of the function in factored form.

Question

in exercises 19 and 20, determine whether ( f(x) = 0 ) has any repeated real solutions. explain your reasoning.
19. graph of a function
20. graph of a function

21. modeling real life archaeologists discovered several huge concrete blocks at the ruins of caesarea. one of the blocks has a volume of 90 cubic meters and the dimensions shown. find the value of ( x ). (see example 3.)
diagram of a rectangular block with dimensions ( (2x - 5) ) m, ( x ) m, ( (2x - 4) ) m

22. modeling real life rivergate tower is the seventh - largest building in tampa. the cylindrical tower has a volume of about ( 56,000pi ) cubic meters and has the radius and height shown (in meters). find the radius and height of the tower.
diagram of a cylinder

23. problem solving you are designing a marble basin that will hold a fountain for a city park. the sides and bottom of the basin should be 1 foot thick. its outer length should be twice its outer width and outer height. what should the outer dimensions of the basin be if it is to hold 36 cubic feet of water?
diagram of a rectangular basin with 1 ft thickness, outer length ( 2x ), outer width ( x ), outer height ( x )

24. how do you see it? the graph of a quartic function ( f ) is shown.
 a. what are the real zeros of ( f )?
graph of a quartic function
 b. write an equation of the function in factored form.

Accepted Answer

### Exercise 19
# Explanation:
## Step1: Identify root behavior
A repeated real solution means the graph touches (does not cross) the x-axis at that point, i.e., a tangent intersection. The graph crosses the x-axis at all its x-intercepts, so no repeated roots.
# Answer:
No, $f(x)=0$ has no repeated real solutions because the graph crosses the x-axis at all intercepts, with no points where it touches and turns around.

---

### Exercise 20
# Explanation:
## Step1: Identify root behavior
A repeated real solution occurs where the graph touches the x-axis (tangent, does not cross). The graph touches the x-axis at $x=3$, so this is a repeated root.
# Answer:
Yes, $f(x)=0$ has a repeated real solution at $x=3$, because the graph touches the x-axis at this point and turns around, indicating a repeated root.

---

### Exercise 21
# Explanation:
## Step1: Set up volume equation
Volume of a rectangular prism is $V = l 	imes w 	imes h$. Substitute $V=90$, $l=x$, $w=(2x-4)$, $h=(2x-5)$.
$$x(2x-4)(2x-5) = 90$$
## Step2: Simplify the equation
First expand $(2x-4)(2x-5) = 4x^2 - 18x + 20$, then multiply by $x$:
$$4x^3 - 18x^2 + 20x - 90 = 0$$
Factor by grouping:
$$2x^2(2x-9) + 10(2x-9) = 0$$
$$(2x^2 + 10)(2x - 9) = 0$$
## Step3: Solve for x
$2x^2 + 10 = 0$ has no real solutions. Solve $2x-9=0$:
$$x = \frac{9}{2} = 4.5$$
# Answer:
$x=4.5$

---

### Exercise 22
# Explanation:
## Step1: Set up volume equation
Volume of a cylinder is $V = \pi r^2 h$. Substitute $V=56000\pi$, $r=x$, $h=7x$.
$$\pi x^2(7x) = 56000\pi$$
## Step2: Simplify and solve for x
Cancel $\pi$ from both sides:
$$7x^3 = 56000$$
$$x^3 = 8000$$
$$x = \sqrt[3]{8000} = 20$$
## Step3: Find height
Height $h=7x = 7(20)=140$
# Answer:
Radius: 20 meters, Height: 140 meters

---

### Exercise 23
# Explanation:
## Step1: Define inner dimensions
Outer dimensions: length $2x$, width $x$, height $x$. Subtract 2 ft from length/width (1 ft from each side) and 1 ft from height (only bottom is thick):
Inner length: $2x-2$, inner width: $x-2$, inner height: $x-1$
## Step2: Set up volume equation
Inner volume (water capacity) is 36:
$$(2x-2)(x-2)(x-1) = 36$$
## Step3: Simplify and solve
Factor $2x-2=2(x-1)$:
$$2(x-1)^2(x-2) = 36$$
$$(x-1)^2(x-2) = 18$$
Test positive real solutions: $x=4$
$$(4-1)^2(4-2)=9	imes2=18$$
## Step4: Find outer dimensions
Length: $2(4)=8$ ft, Width: $4$ ft, Height: $4$ ft
# Answer:
Outer length: 8 ft, Outer width: 4 ft, Outer height: 4 ft

---

### Exercise 24
#### Part a
# Brief Explanations:
Real zeros of $f$ are the x-values where the graph intersects the x-axis. The graph touches the x-axis at $x=-1$ and $x=1$, so these are the real zeros.
# Answer:
The real zeros of $f$ are $x=-1$ and $x=1$.

#### Part b
# Explanation:
## Step1: Identify factors from zeros
Repeated zeros at $x=-1$ and $x=1$ (graph touches x-axis, so even multiplicity). The function is quartic (degree 4), so each zero has multiplicity 2.
## Step2: Write factored form
General factored form: $f(x)=a(x+1)^2(x-1)^2$. Use the y-intercept $(0,4)$ to find $a$:
$$4 = a(0+1)^2(0-1)^2$$
$$4 = a(1)(1) \implies a=4$$
## Step3: Final factored equation
$$f(x)=4(x+1)^2(x-1)^2$$
# Answer:
$f(x)=4(x+1)^2(x-1)^2$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Exercise 19

Step1: Identify root behavior

Step1: Identify root behavior

Step1: Set up volume equation

Step2: Simplify the equation

Step3: Solve for x

Answer:

Exercise 20